00:01
In this problem we are given three matrices ab and c as follows.
00:07
A is equal to 1101 so it's a 2 by 2 matrix.
00:15
B is equal to 1 2 3 in the first column 3 minus 2 0 in the second column so it's a 3 by 2 matrix and c is equal to 1 to 0 minus 1 1 1 3 set 2 by 3 matrix.
00:36
We are going to do some matrix operations with these matrices in three parts of this problem.
00:46
First part, we are going to compute the following matrix products if they exist.
00:53
The first one is b times a times c.
00:57
Now let's check.
00:58
Since we have different dimensions for these different sizes for these matrices, let's check if the sizes are compatible.
01:07
So b is 3 by 2, a is 2 by 2, and c is 2 by 3.
01:16
So in these pairs of, let's say, sizes, like rows and columns, rows and columns, etc.
01:27
We have consistency, so the resultant matrix will be 3 by 3.
01:34
So it exists certainly now we are going to compute it let's copy down these matrices for the sake of not wasting time by writing them again and this is the last one okay the matrix product is independent of the order i mean it doesn't matter if you first compute this product or this product and you my conventions at least i do the products on the right first as i said it's just a conversion maybe just a habit so we have the b matrix over here and let's carry out the product a times c we get 3 2 minus 1 minus 4 3 here and if we compute this final product we will get 9 9 to 9 minus 40 minus 3, 13 to 12, which is a 3 by 3 matrix as expected.
03:11
Now let's consider the other product, namely c times a times b.
03:19
Let's write down the sizes again to see if they are compatible with each other to carry out this product.
03:27
C is 2 by 3.
03:29
A is 2 by 2 and b is 2 by 3.
03:35
So even though a times b exist because their rows and columns are compatible, c and a are not compatible to carry out this product.
03:49
Therefore this product does not exist.
04:00
Okay, this is the first part of the question.
04:03
Now let's move on to the second one part b we have this a matrix again let's emphasize the sizes this 2 by 2 a matrix and we will express it using a product of two matrices x which is 2 by 3 times y which is 3 by 2 2 so we can make this we can express a as a as a as a as the product of two matrices, but x and y will not be unique.
04:44
So that's why in the problem it says find a 2x3 matrix x and 3 by 2 matrix y.
04:52
We have a lot of freedom to choose our matrices.
04:56
And in fact, i'm going to assume just for the sake of simplicity, that x has the following form with lots of zeros and some identity.
05:10
Hidden here.
05:13
Why not? then we will see if this works.
05:15
If it works, then we can, uh, we can safely say that this expansion is possible.
05:22
I mean, this product is possible.
05:25
Okay, to get why now i need to, uh, i mean, i can't make any further assumptions, but i can write why in terms of six unknown variables like a, b, c, d, f.
05:43
Now let's carry out this product.
05:46
We get 1 .00 -1 -0 -0 -a, b, c, d, e, f.
05:55
Okay, the result of this product will be, so we have 2 by 3 here, 3 by 2 here, so it will be a 2 by 2 matrix so that it will match the size of the a matrix.
06:10
And we get a, b, b, d, e over here.
06:17
Now this should be equal to the a matrix, namely 1101.
06:23
So we see that a is equal to 1, d is equal to 1, okay, not b is equal to 0, and e is equal to 1...