00:01
So in this problem, we're given this matrix a as 1 minus 2, minus 2, 5.
00:08
We're told that we don't know b, we don't know matrix b, but we're going to figure it out.
00:13
We do know that a times b is equal to c, and that c is negative 1, 2, negative 1, 6, negative 9, 3 is the matrix there.
00:25
So we're asked for the dimension of b, in other words, the size of it.
00:29
Well, we know that when we multiply matrices, that we do rows times columns.
00:40
And so we know that we have a times b, matrix a times matrix b.
00:51
Matrix a is a 2 by 2, 2 rows, 2 columns.
00:55
We don't know b, but we know that that's c, and c is a 2 rows by 3 columns.
01:07
Okay, so what happens is this outer terms out here give us the result.
01:15
So that means this is a 3 out here.
01:19
And the inner, the columns here have to equal the number of rows here.
01:25
So that's a 2.
01:28
And so we know then that b is a 2 by 3 matrix.
01:40
Okay.
01:42
Now, i want to write this in symbols.
01:45
Right b is symbols of linear equations well we take a times b so let's write b as it's a two by three so we could write a b c d e f like such now if i do a times b a times b i have matrix 1 negative 2 negative 2 5 my matrix b a b c d e f let's write those a little bit neater there and that's equal to c where c is negative 1 to negative 1 2 negative 1 6 negative 9 3 like such which means i can now write the linear equations right do the multiplication on the left so i take the row 1 negative 2 times the first column ad so that means i have a minus 2d is equal to the first entry minus 1 that same row times the second column so that's b minus 2 e is equal to the next entry 2 going across there again that same row times the last column, so that's c minus 2f is equal to that last entry, minus 1.
03:54
And then take the bottom row times the first column, so that's minus 2a.
04:01
Let's just write them side by side.
04:02
I'm going to write them down here.
04:07
Minus 2a plus 5d is 6, bottom entry here, in the first column.
04:19
Same row times the next column minus 2b plus 5e is minus 9 and then minus 2c plus 5f is 3 now next we're asked to write the augmented matrix that this represents and to solve it into row echelon of form okay so if i think about this for a second let me let me put it here and our columns problems will be a b, c, d, e, f, and are constant.
05:09
Okay.
05:10
Then for the first equation i have a one for a, one coefficient, a minus two for d, so it's a zero for b and c and a zero for e and f, and a constant is minus one.
05:22
Do the same thing for the next one.
05:25
The coefficient on b is one, and on e it's minus two, zero is elsewhere.
05:33
And 2 for the constant over here.
05:36
And then the next one is a 1 for c and a negative 2 for f and a minus 1 there.
05:45
Now the next one, a negative 2 for a and a 5 for d.
05:51
0, 0, 0, and 6 for the constant.
05:56
And then minus 2 here and 5 on e, negative 9 for the constant, negative 2 for c, 5 for f and 3 there.
06:14
And so there is our augmented matrix, which we can now solve.
06:20
Okay? to solve it, i need to get the identity matrix in these first six columns.
06:26
Well, the first three rows are already set up, so i just need to eliminate these negative twos down here.
06:35
So let's take row 4 plus 2 times row 1, row 5 plus 2 times row 2 and row 6 plus 2 times row 3.
06:48
So what are we trying to do? we're trying to make those three entries become zeros.
06:54
Okay.
06:57
So the first three rows are the same.
06:59
1 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0.
07:16
0, 0, 0, negative 2, negative 1.
07:22
And then i get 0 and 0, 0.
07:28
And then i have 5 minus 4, because negative 2 times 2 is 4.
07:35
So that gives me a 1 here.
07:38
0, 0.
07:39
And then i have 6 minus 2 is 4.
07:47
Next row, 0, 0, right? because this would be 0 now by design.
07:54
So i get to here.
07:58
5 minus 4 again is a 1, 0.
08:03
So i have minus 9 plus 4 is minus 5.
08:11
Okay.
08:13
0, 0, 0, 0, 0, 1.
08:17
I have the 0 here because i have minus 2 plus 2 is 0.
08:21
And then i have 5 minus 4 again is 1.
08:26
And 3 minus 2 would be a 1.
08:35
Okay.
08:36
Now, i need to get these three to be 0.
08:44
So we're going to do row 1 plus 2 times row 2, 3, 4, row 2 plus 2 times row 5, and row 3 plus 2 times row 6 now.
09:08
All right.
09:11
So i have 1 .0, 0.
09:15
That entry becomes 0 by design.
09:18
0 .0.
09:20
And then i have minus 1 plus 2 times 4, which is 8.
09:26
So minus 1 plus 8 is a 7.
09:29
0 1, 0 .0.
09:32
This entry now is a 0.
09:34
0.
09:37
And i have 2 plus 2 times negative 5.
09:43
So that's 2 plus negative 10, which is negative 8, 0 -0 -1, 0 -0 -0.
09:51
Right, because this negative 2 becomes 0.
09:54
And i have negative 1 plus 2.
09:57
That's negative 1.
09:59
And the bottom 3 rows are unchanged.
10:12
I am now in row -reduced echelon form.
10:17
Remember, this will be a, this will be b, this will be c, this will be d, this will be e, and this will be f.
10:25
So i can now write the matrix b as 7, negative 8, negative 1, 4, negative 5, 1.
10:38
And there's the matrix b right there.
10:49
So this next one, we're given this set of matrices here.
10:55
So we have this first matrix here, this 3 by 3, plus this other 3 by 3, is equal to this last 3 by 3.
11:03
And we're asked to write this in the form of ax equals u.
11:18
Well, to do that, i need to first add the matrices on the left -hand side, and then get them as a multiple result of multiplying two matrices together.
11:39
Right, where x is going to be the matrix.
11:48
Max is going to look like a, b, c, d.
11:52
It's going to look like that.
11:55
All right, so this means that ax, let's see, let's see, let's do this.
12:01
This means the left -hand side looks like.
12:11
So i have the first entry is 1, or 1 plus 1, which is 2.
12:19
The next entry is 2a plus b, and the next entry is 3 plus c.
12:32
Okay.
12:33
Next entry would be 0, 2b plus 0, and then 2 plus c, and then 2 plus 3 plus 3 .3 is 5.
12:42
And then 4d plus a so this is a plus 4d and then 6 plus 0 is 6 and 1 plus 1 is 2 so that's the left -hand side and that looks like our matrix over on the right -hand side 2 c 4b 0 negative a 5 c 6 okay so let's write the augment matrix now.
14:01
So the augmented matrix is going to have these three over here on the left with, oh, but we're going to have, i can't write the augmented matrix.
14:14
I've got to get it in a x equals u, where you is a column vector, don't i? okay.
14:23
Well, the first entry 2 equals 2 doesn't tell me anything, does it? so what's the next entry tell me? 2a plus b is equal to c.
14:36
And then the next entry says 3 plus c is equal to 4b, and then 0 ,0, that's fine...