00:01
In this problem, we want to find a inverse by writing a, i, where i is the identity matrix, and then row reducing until we get some i, b, and b will be our inverse of a, and we'll check that.
00:21
So to do that, let's just start row reducing.
00:23
So we get that our matrix is 2 -2 -9 -1, 1 -0 -0 -0.
00:32
0 .3, negative 1, 010, and negative 2, whoops, negative 1, negative 2, 1, 0 ,01.
00:46
All right.
00:48
And then to row reduce, what we're going to do is take this first row right here, and we want to start with a 1 in our upper left corner here.
00:58
So we want to divide our first row by two so that we get that one there.
01:03
So we're going to do one half row one to replace row one.
01:16
Okay.
01:17
So one half of row one will be one, one, negative one half, zero, zero.
01:29
All right.
01:29
And then the other two rows are going to remain the same.
01:33
03, negative 1, 010, and negative 1, negative 2, 1, 0 ,01.
01:43
Okay, now our next job is to make sure that all of the entries below this one are 0.
01:49
And so since this bottom one here is not 0, we can use this 1 and add this whole row to this row to make this entry a 0.
01:58
So to do that, we're going to add row 1 plus row 2 and replace, whoops, sorry, row 1 plus row 3, and replace row 3 with that.
02:20
So our new row 3 will be our old row 1 plus row 3.
02:26
Okay.
02:28
To do that, row 1 and row 2 are going to stay the same.
02:32
1 -1 -negative 1 -half 1 -half 0 -0 -3 -negative 1 -0 -0 -1 -0.
02:45
And our last row is replaced by row 1 plus row 3.
02:50
So we get 0, and then 1 plus negative 2 is negative 1.
02:55
Negative 1 -half plus 1 -5 is 1ā2 is 1ā2, 0 plus 0 -0, and 0 plus 1 is 1.
03:04
Okay, now we move to our second column here.
03:08
And we want this middle entry to be a one.
03:12
So to do that, let's divide this second row by three.
03:20
So we're going to do one third of row two, becomes our new row two.
03:30
All right.
03:32
So we get one, one, negative one half.
03:35
Our first and third rows are going to stay the same.
03:41
And 0, 1, 1 1 1ā2, 1ā2, 0 .1.
03:49
And we get 0, 1, negative 1 3rd, 0, 1 3rd, 0.
03:59
Now we want the entries above and below this 1 to be zeros, because we're trying to get this to look like the identity matrix.
04:08
So we can pivot off this one to get the other two rows to be to have zeros above and below that entry.
04:16
So first let's get this top one to become a zero.
04:20
And we can do that by taking negative 1 times row 2, negative row 2, and adding it to row 1 plus row 1 to become the new row 1.
04:39
So we're replacing this row 1 by negative row 2 plus the row 1.
04:47
So that means that we're keeping row 2 and 3 the same.
04:51
So 1, 0 ,0, 0, 1, 3rd, 0, 0, negative 1 1 1 1ā2.
05:05
0, 0, 0, 0, 1 1 1ā, 0, 0 .1.
05:08
And we're replacing row 1 by negative row 2 plus row 1.
05:12
So we'll just write out real quick.
05:14
Negative row 2 is going to be 0, negative 1, 1 3rd, 0, negative 1 3rd, 0.
05:24
That's what we're adding to our row 1 up here.
05:27
Okay.
05:28
So we get 0 plus 1 is 1, negative 1 plus 1 is 0.
05:32
1 3rd minus 1 1 1 half is, so let's write this in terms of 6.
05:37
We have 2 6, and negative 1 half is negative 3.
05:43
We get negative one -sixth, 0 plus 1 -half is 1 -half, and negative 1 -3rd plus 0 is negative 1 -third.
05:56
0 plus 0 is 0.
05:58
All right, we're going to now pivot again off this 1, and we're going to add row 2 plus row 3 to replace row 3, because if we add row 2 to our row 3, this becomes a 0.
06:14
Which is what we want.
06:17
So row one and row two stay the same.
06:25
1 half, negative 1 third, 0, and 0, 1, negative 1 3rd, 0.
06:34
1 3rd, 0.
06:35
All right, and then our row 3 is row 2 plus row 3.
06:41
So we get 0, 0, negative 1 3 plus 1 1 1 1ā2 is negative 1 3...