1
00:00:00,527 --> 00:00:07,098
Let's now compute the determinant using
the previous formula of a 3 by 3 matrix.
2
00:00:07,398 --> 00:00:09,581
So there's my matrix A, let's say.
3
00:00:09,862 --> 00:00:14,827
And, once again, I'm allowed to expand
along any row or column and I'll get
4
00:00:14,827 --> 00:00:17,571
the equivalent result when I compute
this determinant.
5
00:00:17,571 --> 00:00:20,764
And then in conclusion,
depending on whether that number is
6
00:00:20,764 --> 00:00:24,570
zero or not, that tells me whether or not
the original matrix was invertible.
7
00:00:25,756 --> 00:00:30,059
I'm just going to use just that first
general formula we wrote here
8
00:00:30,059 --> 00:00:32,588
for a determinant by expanding along
the first row.
9
00:00:32,588 --> 00:00:36,827
So I take the number 5 and then I
multiply by the
10
00:00:36,827 --> 00:00:42,234
determinant of the corresponding submatrix
which is now 0, 2, -1, 3. And then I go
11
00:00:42,234 --> 00:00:47,972
minus, I alternate signs, so check this
minus -3, goes +3 so we flip that and
12
00:00:47,972 --> 00:00:53,129
then I remove again row 1 column 2,
and I just plug in the determinant here,
13
00:00:53,129 --> 00:00:56,065
1,2,2,3, the corresponding submatrix.
14
00:00:56,065 --> 00:01:00,847
And then lastly I pluck out the 2, the
last entry in that row, and then I
15
00:01:00,847 --> 00:01:03,802
multiply that by the corresponding
submatrix here, that
16
00:01:03,802 --> 00:01:09,740
determinant here 1, 0, 2, -1.
So there's my determinant formula.
17
00:01:09,740 --> 00:01:14,460
And again, just as a friendly reminder,
the determinant of a 2x2 matrix is this
18
00:01:14,460 --> 00:01:18,851
sort of cross multiplication process,
ad-bc.
19
00:01:18,851 --> 00:01:20,912
So let's go ahead and clear this up here.
20
00:01:20,912 --> 00:01:28,163
So we have 5, ad-bc, so this is 0-(-2) so
this is going to be a +2, so 5(2).
21
00:01:28,163 --> 00:01:33,086
So plus 3 times ad, which is
3-(-4) is -1.
22
00:01:33,086 --> 00:01:37,266
And lastly we have +2, ad-bc, is -1.
23
00:01:37,266 --> 00:01:40,580
So we have that here. And let's add
these numbers up.
24
00:01:40,580 --> 00:01:46,420
We get 10-3-2 and that results in 5.
25
00:01:46,421 --> 00:01:51,151
That means that my original matrix A
here, is invertible.
26
00:01:51,151 --> 00:01:54,831
So there's some matrix out there I
could find and multiply by that inverse
27
00:01:54,831 --> 00:01:57,589
matrix and then produce the identity.
28
00:01:58,015 --> 00:02:01,395
Since we haven't done a column yet,
let's expand along column 2 here.
29
00:02:01,395 --> 00:02:05,015
And we will compute the determinant
that way and we see that we also get 5,
30
00:02:05,015 --> 00:02:06,602
but we'll have to do little less
31
00:02:06,602 --> 00:02:08,094
work because of the 0 here.
32
00:02:08,094 --> 00:02:11,729
I just want to note right we have this
checkerboard pattern with the signs.
33
00:02:11,729 --> 00:02:15,233
I'm just going to remind you for
the determinant we have a negative with
34
00:02:15,233 --> 00:02:18,428
that term, a positive and then a negative,
we alternate signs.
35
00:02:18,428 --> 00:02:25,310
Let's now compute the determinant of A
one more time, just expanding along this
36
00:02:25,310 --> 00:02:31,753
time column 2. Ok, so I have, negative -3
which is +3 times the determinant of
37
00:02:31,757 --> 00:02:33,917
the submatrix, the same as it was before.
38
00:02:33,917 --> 00:02:39,293
So 1, 2, 2, 3. Ok, now alternate sign +0,
ok I'll just leave this as a place holder
39
00:02:39,293 --> 00:02:42,078
so we can see it. Plus zero. Of course,
that'll go away.
40
00:02:42,078 --> 00:02:46,933
But the submatrix associated with zero
when I remove now column 2 and row 2 is
41
00:02:46,937 --> 00:02:52,027
going to be 5, 2, 2, 3. Ok so that just
goes away. And then I go minus -1
42
00:02:52,027 --> 00:02:57,070
which is plus 1 when I alternate sign,
and then I multiply by the determinant
43
00:02:57,070 --> 00:03:01,061
of the submatrix corresponding with the
-1 here, so I'm going to remove row 3
44
00:03:01,061 --> 00:03:05,947
and column 2 and that leaves me
with 5, 2 and 1, 2.
45
00:03:05,947 --> 00:03:10,418
Let's compute that determinant.
So again, I use the formula
46
00:03:10,418 --> 00:03:20,359
for a 2x2 determinant ad-bc here. Right,
so 3 times 1(3),ad, 3-4 is -1. Ok. plus 0,
47
00:03:20,359 --> 00:03:26,020
and then plus 1 times 10-2 is going to be
8. So what I have when the dust settles
48
00:03:26,020 --> 00:03:31,182
is -3+8 sure enough results in 5.
Point being the determinant can be
49
00:03:31,182 --> 00:03:36,777
ascertained by expanding along any
row or column we get a consistent result.