Determinants I
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An introduction to the determinant of a square matrix. This is perhaps the single most important function in mathematics! It is the basis for understanding invertibility of matrices, scaling factors of linear transformations, and volumes of parallelipipeds. We lay out quite a few properties of the determinant, and look closely at the 2 x 2 and the 3 x 3 cases.
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hello everyone I'm Norman Wapner and
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today I want to introduce the important
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ocean of determinants to central
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theoretical concept in linear algebra
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and well there's also a lot of
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computational aspects to it and it's
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connected with a lot of different things
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and sometimes it's a little bit too
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tricky to give a completely logically
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coherent account of determinants usually
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in linear algebra courses especially
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sort of at the first year level there's
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a bit of cheating involved and I'm
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afraid I'm going to have to be guilty of
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that a little bit here too because
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otherwise it would take a little bit too
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much time to develop some actually some
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relatively serious algebra and even some
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cognitive torques to to lay out the
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whole story completely but at least I'm
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going to be reasonably honest and I'm
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going to you know point out where the
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difficulties are so what is a
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determinant ok so that's a good starting
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point so I matrix a which is square say
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n by n has associated to it number okay
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call the determinant so the determinant
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of a it's expressed in a number of
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different ways sometimes we write
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determinant of a with a bracket
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sometimes we write determinant a without
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brackets sometimes we write a with a
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absolute value around it okay so it's a
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number now what's important about this
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number well it has a number of
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interpretations before I tell you what
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it actually is which is a little bit
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tricky let me tell you you know why
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we're interested in this number this
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number is important for for several
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reasons
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for several reasons the one that
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connects
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most directly with what we've been doing
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is that it turns out that a is
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invertible precisely when the
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determinant of a is not equal to zero so
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they determine it in a kind of a test
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which tells us when matrix is invertible
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another important reason is that the
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determinant of a can be interpreted as a
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scaling factor as a scaling factor when
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a is viewed as a linear transformation
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now I'm just writing this down here
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because we don't know what a linear
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transformation is yet so we're going to
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explain this later in our second course
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okay but that will turn out to be an
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important aspect of it that it's a it's
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a scaling factor it tells us how much a
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transformation scales areas or volumes
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and a third important reason which is
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connected with this second one is that
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the determinant of a captures the signed
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volume of the n-dimensional
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parallelepiped parallel piped vilela Pie
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head okay whose sides are say the
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columns of a so in other words the
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determinant has something to do with
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side bonds
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there's three kinds of at least initial
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sort of motivations for introducing
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deterrents first of all has something to
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do with invertibility and has something
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to do with scaling or how much areas are
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transformed when you think of a as being
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a transformation and we'll we'll learn
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about that later and it also captures
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the signed volume of a parallelepiped
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which is a kind of an n-dimensional
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analog of a parallelogram now that last
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one even though at higher dimensions
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it's actually quite complicated at least
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in two dimensions it's quite
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understandable and that's a really good
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place to start so I want to talk about
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that interpretation now in terms of
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areas in the plane alright so let's look
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at the 2-dimensional the 2-dimensional
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case and signed areas so to do that I
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better define what the determinant of a
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2 by 2 matrix is so if a is say the
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matrix a B C D then the determinant of a
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is ad minus BC that's the definition
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okay that's a definition of the
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determinant of a 2 by 2 matrix and we've
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seen that before we saw that we had a
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formula for the inverse and it was
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exactly this quantity that appeared in
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the denominator of that formula so if
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that denominator is zero then the
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formula doesn't work ok and we don't get
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an inverse so that's so natural that
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that this connects with invertibility
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but I want to explain to you why it
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connects with areas so let's cook up two
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vectors which are say the columns ok
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we'll think about the columns of this
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matrix so here is a vector let's call it
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the vector that's AC and here is another
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vector it's the vector the D
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and what I'm claiming is that if you
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look at the parallelogram that's
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generated by these two vectors it's a
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parallelogram so that means that this
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point up here let's say that's the point
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a and this is the point B this is the
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point C C will be the point with
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coordinates a plus B and C plus D so the
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claim is that the determinant of a this
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ad minus BC is essentially the signed
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area of this parallelogram so area here
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a signed area is equal to ad minus BC so
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I want to convince you of that ok so I
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want to convince you that this purely
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algebraic formula has something to do
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with areas and it's a very different
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kind of area formula from what you're
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used to perhaps in high school because
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it doesn't you know have lengths widths
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exactly in it so we could try to use
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some trigonometry and introduce angles
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and so on but that turns out not to be
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really the best way the best way perhaps
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is to well there's different ways but
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let's do a kind of a simple naive kind
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of thing and sort of look at various
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triangles and rectangles that are formed
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in this picture okay so if we want to
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get at that red area what are we gonna
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do so let me let me prove this claim
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okay proof let's call the the red area
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let's call it Delta all right that's the
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area that we're interested in okay so
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what is Delta well the way I've drawn
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the diagram it's difference between the
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big enclosing rectangle and various
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smaller ones in fact I could kind of
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make it more precise by drawing these
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things here so you can see that what
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I've done now is i decomposed this big
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rectangle into the parallelogram were
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interested in and then there's a couple
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of squares and
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there's also four triangles okay so the
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area of the parallelogram is going to be
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the big rectangular area which is the
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product of color which is the product of
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the base a plus B times the height C
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plus D and I have to subtract some
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things okay
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so let's subtract how about with this
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one here what's the area of this little
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rectangle well the the base because
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we're going from x coordinate of a to a
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coordinate of a plus B it's going to be
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B sorry it's going to be B and the
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height is just C so that's B C and now
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let's also subtract the height of this
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little rectangle up here okay its
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x-coordinate is going to be that's BD
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saw b b and it's y coordinates can be
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the difference that's C okay and then
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there's these four triangular things now
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the triangles are things so this one
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here for example and this one here are
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going to be the same because they are
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they're both have this hypotenuse which
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is this vector here which these two
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vectors are the same so the two
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triangular pieces in green form a
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rectangle okay so the rectangle is going
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to have side the bottom is a and the
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height is C and then there's another one
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what's doing
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let's say so this triangle and this
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triangle here they don't look very much
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the same but they're just two sides of a
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rectangle you put them together there
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they're gonna be the same to give a
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rectangle with dimensions B D okay so we
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do all of that what do we get a we
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expand the first thing out we get AC the
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AC cancels with a see we get a B C that
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cancels with a B C we get an ad which
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doesn't cancel with anything we get a BD
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which cancels with a BD and we get still
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a minus BC left over so that little cut
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and paste shows us or justifies this
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claim that this determinant is actually
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correspond to a signed area so this
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makes it look like the sign area formula
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you know is actually a consequence of
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Euclidian formulas and base times like
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but actually it turns out that when you
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start looking at the theory of area a
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little bit more carefully and generally
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you realize that this is actually a more
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fundamental formula then the base times
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height formula that you're used to
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to define the areas of rectangles okay
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so there's some claim okay that this is
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actually a more fundamental notion of
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area now that takes some explaining but
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if you're are interested in that sort of
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the foundations of that have a look at
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my algebraic calculus 1 course which is
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all about calculus and well it's
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calculus in a very general kind of way
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and hopefully precisely and you'll see
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there that the entire theory of area
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because the integral calculus is all
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about area can be sort of framed in a
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discussion that starts with this notion
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of areas this has turns out to be the
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fundamental notion of area that we can
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use to systematically build up a
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coherent theory of areas of curves so
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yeah it's a very powerful foundational
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tool now to go to three dimensions okay
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I will just show you the picture in
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three dimensions
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in three dimensions there's also a
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corresponding you know formula so I have
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to tell you what the corresponding thing
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isn't it's a bit more complicated than
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this two dimensional line okay so let's
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say if a is a three by three matrix
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let's write it as a b c d e f g h i then
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what is this magical determinant so
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here's a formula for the determinant of
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a it is squeezed six terms a times e
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times I minus a times F times H plus B
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times F times G minus B times D times I
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and plus C times D times H minus C times
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e times G you stare at this you'll see
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that there are six terms with each each
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containing a single entry from each row
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and column so for example if we you know
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look it's a b FG that's that term times
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that term times that term and indeed
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those three terms is one in each row and
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one in each column and all these six
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terms all have exactly that property and
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in fact these are exactly all of the
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terms that you can cook up from the
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entries that that are formed by taking
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products of elements with exactly one in
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each row and in each column and it's not
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too hard to see that there's going to be
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a general three factorial of those so
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there's three factorial terms these are
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the number of ways of choosing one
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element from each row and column of the
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matrix and so this is a quite an
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imposing formula
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let's get rid of this thing so let me
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put a box around it so it's a very
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imposing formula perhaps but it's also
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quite elegant but it does show you that
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no calculating determinants is going to
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be potentially complicated because this
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is only the three-dimensional version
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right so the higher dimensional versions
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are more complicated but I do want to at
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least give you an impression of what
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that you know how you should think about
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what this number means okay so here is
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an interpretation and interpretation
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we're in three dimensional space let's
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look at that three columns of the matrix
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a ok so those are three vectors maybe
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there's the vector a what was a b c d e
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f g so a d g that's one vector another
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vector is the second column b e:h BPH
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and then there's a third vector which is
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the third column c fi okay c fi now
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these three vectors determine a
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parallelepiped which is three
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dimensional analog of a parallelogram so
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let's say we take two of them say these
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two they form a parallelogram for take
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these two we form a parallelogram we
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form these two we take a parallelogram
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okay and then we can sort of fill this
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thing out to something that looks like
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this this is a parallelepiped parallel
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pipette whose sides are all
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parallelograms grams
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and the claim is that the determinant of
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matrix a corresponds to the signed
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volume of this parallelepiped so we're
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talking about the volume
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three-dimensional volume of this box
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this box with slanted sides determined
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by these three vectors and again if
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you're you know gonna go through and
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develop a theory of area or volumes in
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three dimensional sense it's actually
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quite subtle okay and it turns out right
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that this is actually the starting point
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which is best it's best to start with
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this as your notion of science and then
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show that this notion gives rise to a
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consistent framework of volumes that
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that captures the intuitive properties
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that we would expect volumes to have but
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it is that's that's a fair amount of
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work okay that's a fair amount of work
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highly non-trivial but I do want you to
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have this idea in mind that determinants
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at least in two dimensions and in three
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dimensions represent volumes you might
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also ask just for completeness what
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about the one-dimensional case we should
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also say that so in the one-dimensional
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case suppose we have just emmaus a with
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a single entry well then determinant of
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a it's just the single entry a by itself
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and in some sense that's also a kind of
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a measure of extented system it's like a
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one-dimensional area or an extent so
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here's the origin here's a point a and
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we're talking about this this vector
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okay it's a vector a okay and this
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number represents how big it is but in a
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signed way not in an absolute value
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sense more sense so it's not like a
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distance so much as a displacement okay
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so it can be positive or negative
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depending on
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the value of a okay great so what what
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is the the official definition of
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determinant so the way we're going to to
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do this is to use an inductive procedure
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so that at least we have a definition
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and the deductive procedure is sort of
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consistent with the three cases that
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I've shown you now okay so the
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definition is for the N dimensional and
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dimensional determinant is it's an
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inductive definition okay so if a is n
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by n matrix say with let's give the the
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elements some name so let's say that a
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is say a matrix consisting of a IJ
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little a IJ so that's the IJ entry of
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this matrix tagging okay then then let's
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let a sub IJ capital a sub IJ okay be
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the n minus 1 by n minus 1 matrix
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obtained from a by removing the 8th row
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and J column of a so this is called a
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minor a minor of a ok so that's an n
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minus 1 by n minus 1 matrix it depends
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on I and J then our definition is that
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the determinant a it
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a11 times the determinant of a1 1 minus
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a1 2 times the determinant of a1 2 plus
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and then alternating plus and minus
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until you get to the enth 1 the nth one
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will be a1 n times the determinant of a1
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n with a minus 1 to the end in front of
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it so alternating plus or minus
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depending on whether and is even or odd
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that is an inductive definition and
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maybe I should say the determinant of a
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which is also equal to an absolute
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values is given by the following so this
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is inductive because the N by n matrix
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depends on evaluating n n minus 1 by n
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minus 1 matrices so an N by n
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determinant depends on n n minus 1 by n
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minus 1 determinants and so you can see
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that as n gets larger the computational
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problem of calculating determinants from
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this definition is going to be a lot a
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lot of work so let's let illustrate this
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definition ok so you can see that it
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corresponds with what you know I was
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doing earlier so let's apply this to the
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example that we did before but now we're
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going to use this determinant so we'll
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stick with the same 3x3 ABCDEF g h i
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according to the definition now
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according to the definition what are the
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determinant of a be it should be 8
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time's the determinant of the matrix e F
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H I so notice that what I'm using here
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so this what's inside here okay is the
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matrix a sub 1 1 it's what we get when
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we remove the first row and the first
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column it's that little two by two
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matrix therapy obtained by removing the
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row and the column of the 1 1 entry
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which was a so that's the first element
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in this sum the next element is moving
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along to the second the second entry
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which is B and then we have to multiply
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by the determinant of the matrix that we
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get by removing the row and column
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through B that'll be the 2x2 matrix D F
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G I so that's the minor corresponding to
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1 to the entry fee and then plus C the
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last term will be times the determinant
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of the 2x2 matrix d e G H all right so
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this is the minor a 1/2 this is the
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minor a 1/3 so we're taking a linear
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combination of these three minors each
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of which is itself a two-by-two
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determinant
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ok and that linear combination gives us
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the entire determine so let's see what
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happens when we do that
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so this two-by-two determinant is e i- f
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h this next one is d i minus GF and this
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next one is plus C times D H minus e G
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ok and if we expand that we get a e i- a
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f h- b d oh maybe I'll do the plus first
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plus b g f- b di plus c th minus
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CEG and hopefully that's the same as
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what I originally told you was the
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definition up here okay so this has been
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an inductive way of seeing how to go
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from the 2-dimensional to the
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three-dimensional and you can check also
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that this induction goes from the
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one-dimensional to the two-dimensional
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but it also at least then in principle
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allows us to calculate higher
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dimensional determinants so to calculate
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a four by four determinant we would have
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to calculate 4 3 by 3 determinants and
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take the corresponding combination so we
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have to talk about the properties of the
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determinant properties of the
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determinant so first of all the
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determinant of the identity matrix is
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always equal to 1 the determinant of the
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identity matrix is also always equal to
27:20
1/2 if you interchange or swap two rows
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or two columns then the determinant
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changes sign changes sign in other words
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it becomes the negative of what it was
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before interchanging two rows you
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multiply the determinant by minus 1 or
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interchanging two columns you multiply
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the determinant by minus 1 that was an
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elementary row operation right so what
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about the other elementary row
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operations third if you multiply
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row by lambda then the determinant is
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multiplied by lambda fourth if you add
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multiple of one row to another the
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determinant does not change another
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important property of the determinant is
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that the determinant of the transpose of
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the matrix equals the determinant of the
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matrix taking the transpose of the
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matrix doesn't change the determinant
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and I want to mention one more property
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which is related to the ones I mentioned
29:54
before about you know it's sometimes
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called a multi linear property it's not
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exactly a linearity in the obvious sense
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it's a multi linearity that I will write
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it like this if you have a matrix a okay
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and say has first column is say v1 plus
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v1 prime that's the first column and the
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second column is v2 and then the second
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and then antha column is VN okay
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if so if a is a matrix whose first
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column is a sum of two vectors and but
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the other columns are just individual
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vectors then the determinant of a equals
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the determinant of the matrix with just
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v1 and all the other columns the same v2
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up to VN plus the determinant of the
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matrix that has the one Prime v2 all the
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way to the end so this is linear so the
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determinant is linear in in the first
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column and in fact it's linear in any
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column because the swapping property
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above shows that that's going to happen
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so the determinant is linear in any
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column or in any row as it turns out or
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in any row because the transpose has the
31:54
same determinant okay so that's sort of
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a bunch of properties what about sort of
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like proving or understanding these
32:04
properties well the thing is our
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definition is is complicated it's not so
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easy to prove all of these things in in
32:15
general but we can prove them or
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demonstrate them for the two by two case
32:19
and a three by three case and for most
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applications you know it having to do
32:25
with ordinary geometry and in three
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dimensional space two by two and three
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by three determinants are you know what
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you meet a lot of the time so it's
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actually nice to have you know direct
32:38
verifications of the two dimensional in
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the three dimensional properties
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directly because we have these very
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explicit formulas for the determinants
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of two by two and three by three
32:48
matrices so that you can see the various
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properties so for example here is the
32:54
3x3 property of the determinant and if
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we look and think about how does this
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you know how does this dependent on the
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first column a D and G then you see that
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the a D and G there's a here here
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there's a G here the ad does a D here
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it's a D here there's a G each one of
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these terms has exactly one element from
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that first column so this is the reason
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why this thing is linear in the first
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column so if we you know we say
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multiplied the first column by two okay
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then all of these entries would multiply
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by doing the entire determinant would
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multiply by two or if we replace this
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first column with the sum of two columns
33:51
then the entire determinant would be the
33:54
sum of the two corresponding determines
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which makes me think I should probably
34:00
also this multi linear property that's
34:04
that's from one part of it
34:06
the other part is also that if I take
34:12
the determinant of say C times v1 so if
34:18
I multiply the first column by say C and
34:23
take the determinant that what I'm going
34:25
to get is going to get C times the
34:27
determinant of the original matrix B 1
34:30
through the end and this works for any
34:38
column works for any column and any row
34:50
consequence of that which is actually
34:52
quite useful is that if you multiply the
34:55
entire matrix by C the determinant of C
35:00
times the entire matrix okay
35:04
that means you multiply the first column
35:06
by C you're multiplying the second
35:07
column by Cu month like the ads column I
35:09
see each one of those individually is
35:11
going to multiply the determinant by a
35:14
factor of C so all together you're going
35:17
to be multiplying the determinant by C
35:22
to the N so the determinant of C a is
35:27
equal to C to the end determinant a when
35:30
a is n by n okay so those are some sort
35:41
of abstract properties in our next
35:44
lecture we'll talk a little bit more
35:46
about some of those properties and
35:47
verify some and get more experience with
35:51
actually calculating determinants so
35:54
that you can see these various
35:55
properties in action