Determinants III (some properties)
3.01K subscribers
SUBSCRIBED
We explain some basic properties of the determinant of a general n by n matrix, culminating with the multiplicative property and the condition for invertibility.
00:00
hello everyone i'm norman robert today i
00:03
want to carry on with the lecture that
00:06
we were talking about last time about
00:07
determinants and also of an alternate
00:10
count oriole interpretation and in fact
00:14
i just got around to almost finishing
00:17
this problem here we were looking at
00:19
this diagonal matrix and I was
00:20
convincing you that the determinant is
00:23
just actually the product of the
00:24
diagonal elements which in this case is
00:27
4 times 3 times 1 times 7 which is 12
00:31
times 7 which is 84 okay so there's a
00:35
generalization of this we might write it
00:37
as a proposition that if a is n by n
00:45
diagonal matrix then determinant of a is
00:56
the product is the product of the
01:04
diagonal entries and the argument is
01:10
just the same as I've just shown you in
01:13
that 4x4 case there's a generalization
01:16
of this to what's called an upper
01:19
triangular matrix so I'll just write it
01:23
down proposition if a is an N by an
01:28
upper triangular matrix and that means
01:37
ie that means that the IJ entry the IJ
01:43
entry of of a thing in the eighth row
01:49
and jth column is equal to 0 if I is
01:55
bigger than J so if a and by an upper
02:02
triangular matrix which has this
02:03
property then again determinant of a is
02:08
the product
02:12
of the diagonal entries so for example
02:25
suppose we have the matrix a which is 3
02:28
minus 1 2 4 7 5 and it's zeros below the
02:39
diagonal ok so like this entry here this
02:43
is the entering the second row first
02:46
column it's the 2 1 entry and that's 0
02:50
this one here that one is the between
02:55
the 3 1 entry that's also 0 this one
02:58
here the 3 to entry is also 0 ok so this
03:03
matrix here to find its determinant
03:06
using the same kind of argument that we
03:09
had before that we're basically looking
03:11
to place rooks on this 3x3 chessboard so
03:15
that we get nonzero product of entries
03:19
there's only one place to put the rooks
03:21
because in the last row for example
03:25
there's only one place where you could
03:26
put the rock and then if you put the
03:29
rook there in the second last row
03:30
there's only one place where the rook
03:31
can go in which is that one and in the
03:34
first row there's only one place where
03:36
the brook can go so those are the only
03:38
positions possible for 3 rooks so that
03:43
they're all sitting on nonzero values
03:46
and so therefore in this case so the
03:50
determinant of a is just equal to 3
03:54
times 4 times 5 which is 60 now this
04:00
this is very important because it
04:03
connects directly with row reduction
04:06
remember that when we row reduce a
04:09
matrix we end up getting something in
04:11
row echelon form and then ultimately in
04:13
reduced row echelon form so note that a
04:20
square matrix
04:24
in row echelon form is always offer a
04:36
triangular so this proposition applies
04:44
and tells us that once we have something
04:46
in row echelon form its determinant is
04:48
just the product of the diagonal entries
04:50
let me remind you for example if we have
04:53
a four by four situation maybe once one
05:01
there may be two there and then one here
05:05
three zero one four okay and one more
05:17
row of zeros so here is a four by four
05:23
matrix which has been reduced into a row
05:26
echelon form the leading entries are all
05:29
staggered successively to the right as
05:33
we go down in particular they all have
05:36
to be in this sort of top diagonal half
05:39
of the matrix everything below that
05:42
diagonal is necessarily zero if it's in
05:45
row echelon form now in this case here
05:49
it's actually we can see that the actual
05:52
diagonal consists of the elements 1 0 0
05:55
0 it's in row echelon form and its
05:58
determinant is necessarily a 0 so here
06:01
the determinant equals 1 times 0 times 0
06:05
times 0 reading down this diagonal if
06:07
the diagonal there and that's of course
06:09
0 okay so those are useful properties
06:18
let me give you a few more helpful rules
06:25
another proposition if a is an N by n
06:30
matrix with Row
06:37
or column of all zeros then determinant
06:51
of a has to be zero so if you have a
06:58
matrix and there is a say a row of zeros
07:03
somewhere it's a five by five matrix
07:05
okay it's got something here and it's
07:08
got something here we don't really care
07:09
what those things are it's a five by
07:14
five matrix but it has a row of zeros
07:16
somewhere it doesn't matter where or a
07:17
column of zeros then we can conclude
07:22
that the determinant of a is zero and
07:26
why is that well it's because in the
07:30
main proposition that I told you way up
07:33
here this theorem game this is the
07:35
alternate kind of permutation form for
07:38
the determinant we saw it's a sum of
07:40
terms each of these terms is a product
07:43
of elements where we're taking elements
07:46
one to each row this this thing here is
07:50
something in the first row that's in the
07:52
second row that's in the nth row so if
07:54
there's one element being taken from
07:55
each row to form each one of the
07:58
products which are then summed up to get
08:01
the entire determinant with these plus
08:03
or minus one signs in front okay now if
08:07
we go back to our situation here every
08:11
such sum will correspond to some
08:14
position of rooks like maybe this one
08:20
there you know there was five rooks on
08:23
this five by five chessboard none of
08:24
them are taking each other and we see
08:27
necessarily that one of those rooks has
08:30
to be in that row that consists of all
08:32
zeros so every rook configuration has
08:36
this property so all of the terms all of
08:42
the terms in the in the determinant
08:52
we'll have a zero factor coming from the
09:03
entry in that row in that in this case
09:10
third row so this is a very quick way of
09:16
telling that matrix is going to have
09:20
determinant 0 it means it's happening
09:24
when there's a row or column of zeros
09:26
and another proposition that's related
09:29
to this one proposition that if two rows
09:38
or two columns of n by n matrix a are
09:50
identical then determinant of a equals 0
10:03
so for example same 3x3 case here's a
10:07
matrix a say zero four three one two
10:16
five one two five we see that these two
10:22
rows are equal therefore we can conclude
10:28
without any calculation that the
10:31
determinant of a is zero and why is that
10:36
well the reason is because reason adding
10:42
a multiple of one row to another doesn't
10:45
change the determinant that's one of our
10:48
elementary row operations and we know
10:51
how it effects the determinant it
10:52
doesn't change it so adding a multiple
10:55
of
10:59
row to another doesn't change the
11:09
determinant but then we can just
11:16
subtract one row from another so
11:20
subtracting one row from another we get
11:30
our row of zeros and hence from the
11:39
previous proposition the determinant is
11:41
going to be zero okay that's all good
11:49
and now I come to the kind of most
11:53
important fact about determinants in
11:59
terms of you know properties so we're
12:03
going to call this on a theorem and it
12:06
is that if a and B are n by n matrices
12:14
then determinant of a times B the
12:22
determinant of the product of the two
12:23
matrices is equal to the product of the
12:28
determinants equals determinant a times
12:33
determinant B this is a really beautiful
12:41
a beautiful fact okay and it's really
12:43
this fact primarily which is responsible
12:46
for the importance of determinants
12:48
ultimately it's somewhat a non-trivial
12:52
fact okay yeah it's it's a little bit
13:01
tricky not at all obvious let me give
13:04
you an example so let's choose some
13:07
matrices let's say two-by-two say 2 1 3
13:10
5
13:11
let's say B equals 1 0 minus 1 4 ok then
13:17
what's the product a B let's quickly do
13:19
the product 2 times 1 plus 1 times 0 is
13:22
2 and 3 times 1 plus 5 times 0 is 3 and
13:26
minus 2 plus 4 is 2 and minus 3 plus 20
13:32
minus 3 plus 20 is 17 ok so let's
13:37
calculate the determinant determinant of
13:39
a is 2 times 5 minus 3 times 1 that's
13:44
equal to 10 minus root of 7 okay now
13:47
determinate of B that's actually an
13:50
upper triangular matrix so the
13:51
determinant is the product of the
13:53
diagonal elements 1 times 4 which is 4
13:55
and over here the determinant of a B is
13:59
2 times 17 minus 3 times 2 which is 34
14:05
minus 6 that's 28 and yes indeed 28 does
14:10
equal 7 times 4 that's a very beautiful
14:24
fact and perhaps one more theorem that's
14:28
really important to remember so the
14:30
other crucial thing why determinants are
14:33
important is that n by n matrix a is
14:41
invertible it precisely when determinant
14:48
of a is not equal to 0 so that's how we
14:59
tell whether matrix is invertible
15:01
whether it has an inverse it has an
15:04
inverse precisely when its determinant
15:07
is not equal to 0 and so in particular
15:09
if it's determinant is equal to 0 that
15:11
means it's not invertible you cannot
15:13
invert it now both of these theorems are
15:20
requiring a little bit more in-depth
15:22
look at you know how to compute matrices
15:27
and that's what we'll do in our last
15:29
video we'll talk about you know how to
15:32
compute matrices and the connection with
15:35
row reduction and that will also allow
15:39
us to outline at least the direction of
15:41
the proofs of these two important
15:43
theorems okay but they're very crucial
15:46
theorems that the determinant is
15:48
multiplicative and that the determinant
15:50
captures invertibility turn of a is not
15:54
equal to zero that's exactly the
15:55
condition for invertibility