00:01
For each of the following statements, a and b, we gotta say if the statement is true or false and justify the answer.
00:10
So let's see part a.
00:13
A, b and c are square matrices of the same dimension and a is an invariable matrix.
00:20
Then the power to 2023 of a inverse times b times c times a is a inverse times b to the power 2023 times c to the power 2023 times a.
00:38
So let's see that first.
00:43
Then let's think about smaller power.
00:46
Let's think about the square of that matrix.
00:49
So we have a square times b times c times a square.
00:56
By definition, this is that matrix times the same matrix, that is, this matrix times itself.
01:08
Now we can group the middle matrices a and a inverse, that is, this is a inverse times b times c times a times a inverse, that is, we are grouping these two matrices in the middle, which are one beside the other.
01:29
And we know this product here is the identity matrix because by definition the inverse of a matrix times a matrix is in the identity matrix.
01:38
So we get a inverse times b times c times the identity matrix times b times c times a.
01:46
Now the identity matrix we can group it with b for example and we get a inverse times b times c times b times c times a.
01:58
And here of course if we group these two matrices and these two matrices it is clear that it is a inverse times b times c square, that is the square of this matrix, times a.
02:15
But now we cannot say in general we cannot say this is a inverse times b square times c square times a.
02:31
It's not true because the product of two matrices is not commutative in general.
02:37
That is, v times c is not necessarily equal or in general is different from c times v.
03:44
So we can say that the product of two matrices is not commutative in general...