Elementary row operations on matrices
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We introduce the three basic kinds of row operations on the augmented matrix of a system of linear equations that form the basis for row reduction, or Gaussian elimination.
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hello everyone I'm normal inverter okay
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so we're gonna talk about elementary row
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operations on matrices so this is moving
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towards explaining role reduction or
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Gaussian elimination a systematic
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approach to solving systems of linear
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equations okay so the first thing to do
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is to remind you that we had a matrix
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formulation for a system of equations so
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here for example is system of three
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equations in four variables X 1 X 2 X 3
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X 4 there's three equations and
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correspondingly here below it you see an
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Augmented matrix for that system ok and
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we see just on the left hand side is the
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encoding of all the coefficients that
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appear on the left hand side of the
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equations and then on the right hand
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side here the column 1 2 3 is just a
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column that appearing on the right-hand
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side of this system of equations so the
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idea is that instead of working with the
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equations we're going to work with the
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Augmented matrix instead and the
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advantage of that obviously is that we
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don't have to keep writing down the
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variables X 1 X 2 X 3 X 4 because the
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positions of those variables is already
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kind of clear by the way the matrix
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organizes things okay so we can just
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work with these numbers and then
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interpret what we get in terms of the
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original equations now the strategy is
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that we are going to exploit we use
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three different types three different
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transformations or operations on
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augmented matrices
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and these operations that we're going to
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use have the crucial property that they
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do not change the solution set okay so
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whatever is a solution to the original
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system will still be a solution to any
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of these modified matrices that we're
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going to create and conversely any
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solution of a modified Augmented matrix
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that we get is also a solution to the
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original system so that's the key point
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that we're making transformations that
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don't ultimately change the set of
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solutions to our system so we use three
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different operations on a committed
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matrices and maybe you should say
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matrices that do not that do not change
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the solutions okay and these are our so
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the first one is is interchange of any
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two rows and the second one is going to
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be adding multiple one row to another
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the third one is going to be multiplying
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a row by a nonzero number so these are
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going to be called these are these are
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going to be called elementary elementary
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row operations
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and they're going to be acting on
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augmented matrices for a system we also
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going to introduce some kind of notation
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for for these so let us have an example
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or two so let's look at the first one
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interchange of two rows okay so suppose
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we have some Augmented matrix
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maybe 2 4 3 7 1 0 2 minus 5 and 4 1 half
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3 2 I remind you that the numbers
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involved here don't always have to be
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integers usually we will illustrate
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things with integers but we don't
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actually officially have to and the fact
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later on we'll see that fractions do
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naturally appear ok so what might we
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want to do you might want to interchange
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two rows and our notation is that our
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first row will denote by r1 and our
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second row by r2 and our third row by r3
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okay that's the convention that we'll
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use so now we're going to make a
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transformation and we're going to change
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this to the following 1 1 0 - -5 - 4 3 7
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4 1/2 3/2 ok what have I done here I
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have interchanged r1 and r2 and we'll
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denote that with this little symbol here
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r12 double arrow r2 that's telling us
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that we're interchanging Row 1 and Row 2
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of the original matrix to get this new
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matrix now that's pretty clearly not
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going to change the solutions to this
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system because remember each one of
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these rows corresponds to a linear
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equation in in this case x1 x2 x3
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perhaps or we don't actually know what
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the
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Able's are maybe the variables on x y&z
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but whatever they are the three
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equations that we have at the end in the
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second Matrix is exactly the same as the
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equations that we have in the first
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matrix it's just that they're in a
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different order so it just changed the
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order of the the equations that's not
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going to change you know what the
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solutions are why might we want to do
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this so we might want to do this because
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it's very convenient to have a one in
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the top of column that we're working on
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we're going to be working on kind of
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clearing elements from columns and
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having a one there makes it particularly
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convenient to remove by subsequent row
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operations the entries below that so
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it's a kind of a way of organizing
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bookkeeping the row reduction algorithm
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okay that's why we do this kind of
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things usually to bring more convenient
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entry to the top of some column okay
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it's a relatively simple operation the
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second one is the most used so it is
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adding a multiple of one row to another
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adding a multiple of one row to another
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so let's illustrate that say I'll start
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with that matrix that we just had in the
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previous example 1 0 2 minus 5 2 4 3 7 &
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4 1 half 3 2 and what I'm going to do is
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I'm going to transform this by changing
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the second row and I going to change the
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second row by adding a multiple of the
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first row to it in order to get rid of
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that 2 that's sitting
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they're underneath of the one okay so
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that's what our plan is to get rid of
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that to to make it a zero and way we're
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going to do that is we're gonna actually
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add minus two times the first row to the
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second row and the way we're going to
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write that is in kind of computer
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language so we're going to write our two
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equals this is like the new are two on
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the left-hand side of the equation
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that's late you should interpret that is
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equal to the old are two minus two times
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R one so then we're gonna get a new
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system the first row is not going to
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change but the second row is because
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we're going to subtract twice the first
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row from it so 2 minus 2 is 0 4 minus 2
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times 0 is still 4 3 minus 2 times 2 is
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minus 1 we also have to do it on the
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right hand side 7 minus 2 times minus 5
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he is a total of 17 can the third row
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well we're not doing anything to that so
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we'll leave it where it is okay so let's
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make sure that we understand what this
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notation means here so we have added
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minus 2 times the first row or first row
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are 1 2 the second row are 2 and to
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understand what's really going on there
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so let's have a look at what it looks
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like at the level of equations so at the
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level of equations maybe we'll use say
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using say XY and Santa's are variables
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so the first equation that we originally
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had was X plus 2z equals minus 5
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and then 2x plus 4y plus 3z equals 7 and
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4 X plus 1/2 y plus 3z equals 2 and then
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after we've performed this we still have
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X plus 2z equals minus 5 and now we have
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2x minus 2x is 0 x plus 4y minus 3z
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equals 17 and we still have 4 X plus 1/2
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y plus 3z equals 2 so we performed well
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actually this level it's perhaps safer
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to say that we have just worked on the
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level of the equations so equation 2
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equals the old equation 2 minus 2 times
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the equation 1 that might be a
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corresponding thing in terms of the
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equations rather than rows and we can
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see that this is not going to change the
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solution set why is that so note that
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any vector say XYZ which satisfies the
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first system also satisfies the second
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system because if X Y is that satisfy
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these three equations here then these
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three equations here are obtained from
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these ones by just combining them so
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these ones these new ones over here are
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just going to still be satisfied if x
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y&z cents by the original set but what
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about the converse question how do we
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know that anything that satisfies the
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second equation also satisfies the first
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equation
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so it's sort of the converse so
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conversely we can see that any solution
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of the second system is also a solution
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to the first system because because the
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operation is invertible because the
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operation that we've just done is
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invertible which is another way of
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saying reversible we can undo it how
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would we undo that so how would we go
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back and recreate the first system from
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the second system let's think about that
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I'm going to do that over here in in
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blue all right suppose that we started
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with the system on the right hand side
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this one here and we wanted to go to
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this one over here let's do it in red
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okay what would we do well the first
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equation is the way it is the third
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equation is the weight is only the
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second equation is being changed and
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what actually is happening to the second
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equation the new second equation that is
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the one over here is equal to the old
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second equation plus two times the first
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equation you see that right so if we
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take this equation the second one over
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here and we add twice the first one over
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here
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then we're going to recover the second
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equation over on the original side
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that's explaining how to undo or how to
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invert or how to reverse this second
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elementary row operation and that's the
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reason
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why it's also the case that if we have a
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solution to the second one the one on
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the right hand side then we'll also get
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a solution to the left hand one because
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the left and one is obtained from the
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right hand one by just taking equations
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and adding multiples of one to the other
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so this is a kind of the key the key
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operation this is the one that you do
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like you know 90% of the time really
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okay so we're going to be using this
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operation quite a lot but there's also a
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third one the third operation is to
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multiply a row by a non-zero number okay
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so let's go back to the last matrix that
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we had the 1 0 2 minus 5 1 0 2 minus 5
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and then the next one was 0 4 minus 1
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1704 1 minus 17 then the last one was
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for 1/2 3 2 4 1 half 3 2 okay
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so let us transform this by using this
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third kind of elementary row operation
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and why might we want to do that well
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usually it's to normalize the numbers
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that appear sometimes it's because we
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want to remove fractions sometimes it's
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because we want to create a 1 in a
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particular spot so let's illustrate that
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here because let's have a look at this
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third row which we haven't done anything
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with yet so let's take the third row
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actually we're going to use notation for
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row so Row 3 let's take Row 3 and the
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new Row 3 is going to be 2 times the old
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Row 3
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yeah so this equation that I've just
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written is not really a good
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mathematical equation it's the kind of
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equation that a computer scientist might
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use in those of a programming
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environment so the left-hand side is
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referring to the new Row 3 while the
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right hand side refers to the old Row 3
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and say that the new Row 3 is gonna be
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twice the whole of Row 3 ok so what do
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we get out of this well the first row
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stays where it is and the second row
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stays where it is and the third row is
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multiplied by 2 so 8 1 6 4 so we
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performed that that operation and we
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also note that this is also this is also
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invertible or reversible to reverse it
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we would just multiply by the inverse of
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what we multiplied previously by the
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inverse of the reciprocal and for that
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of course it is important that we're
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agreeing to multiply row by a nonzero
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number you are not allowed you are not
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allowed to multiply a row by 0 that's
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effectively killing that row then you
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losing information it's no longer the
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same kind of system so that's not an
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allowed operation because we want the
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the set of solutions to stay the same
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right in this case it's kind of obvious
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that it stays the same if you just take
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an equation and you multiply both left
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and right hand sides by the same number
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in this case 2 the solutions are not
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changing it just becomes more convenient
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for the algorithm to use that equation
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in that altered form okay so these are
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the three elementary row operations and
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we are now going to then exploit them to
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show how to solve systems of linear
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equations so I'll do that next time