In this segment, we're going to talk about is A times B same as B times A? We just basically
we're trying to see that whether the commutative law of multiplication of matrices is that
valid? So the first thing which you have to understand is that if A times B is possible,
then the number of columns of A has to be the same as the number of rows of B. So let's
suppose this is m by n and this is n by p, yes A times B is possible. Then if I try to
say hey, B times A, this is n rows and p columns and this is m rows and n columns. I already
see that the number of columns here is not the same as the number of rows here. Unless
p is the same as m. So if p is the same as m, then I have to know that hey, then the
size of the matrix B has to be squared, and A also has to be squared and they have to
be the same dimension. Only then is there a possibility that A times B would be the
same as B times A. So we can very well see that if A times B is allowed, B times A cannot
be allowed, and I think we only allowed now if this is the same as this. And if we consider
this to be the same as this, then we have to also assume that it has to be a square
matrix. Both of these have to be squared matrices in order for that to be possible. But even
then, if we have two squared matrices, if I have two squared matrices let's suppose,
of the same size, n by n, and I say hey, is this true? No this is not generally true,
even if you have two matrices which are squared and of the same size, even then you will find
out that the commutative law of multiplication is not valid. Let's take an example. If we
have A given as 6, 3, 2, 5, and B given as -3, 2, 1, 5. So we've got these two matrices
here, and then we find out if A times B is allowed. Because the number of columns here
is the same as the number of rows here. But if we do the matrix multiplication by what
we have learned in matrix multiplication. We get -15, 27, -1, 29. And then if we want to
calculate B times A, do we get the same matrix? I don't know. So let's see what we get for
B times A, B times A turns out to be -14, 1, 16 and 28. And that's all the same matrix,
so in this case A times B is not equal to, B times A. So in most cases, you will find
out that commutative law of multiplication is not possible for multiplication. Commutative
law is not valid for multiplication. So there are two things that are going on. First, you
have to match the orders, in such a fashion that A and B are both of the same size and
they're both squared. And that's the only necessary condition. And by the sufficient
condition then is to multiply two matrices and see whether they are the same element
to element. Which we don't find to be the case in here but there are cases where A times
would be equal to B times A. That'll be more clear when we see some special cases where,
let's suppose this matrix is the inverse of that matrix or this matrix is the inverse
of that matrix. Then that will be true. But that's down the road we're going to look at
that. But this is the end of this segment.