Given the two-by-two matrices 𝐴,
which is equal to eight, negative three, one, negative two, and 𝐵, which is equal
to eight, negative three, one, negative two, is 𝐴𝐵 equal to 𝐵𝐴?
So we’ve been given two two-by-two
matrices. And we’re asked whether 𝐴𝐵 is
equal to 𝐵𝐴. So we’re being asked whether we get
the same result when we multiply the two matrices together in different orders. In general, we know that matrix
multiplication is not commutative, which means that we get a different result if we
multiply the matrices together in a different order. In fact, unless the two matrices
are each square matrices of the same order, it won’t even be possible to find both
Matrix multiplication can be
commutative under special circumstances. For example, if both matrices are
diagonal matrices of the same order, meaning that they are square matrices. And all of the elements that aren’t
on the leading diagonal are equal to zero. Matrix multiplication will also be
commutative if one matrix is the identity matrix. And the other is a square matrix of
the same order.
Now, neither of these conditions
apply here. But if we look at our two matrices
𝐴 and 𝐵, we see that they do have another relationship. 𝐴 and 𝐵 are each two-by-two
matrices. So they have the same order. But more importantly, each of their
elements in corresponding positions are the same. In this case then, matrix 𝐴 is
entirely equal to matrix 𝐵. For this reason, it doesn’t matter
in which order we multiply these two matrices together. The product 𝐴𝐵 will be equal to
the product 𝐵𝐴. And in fact, they’ll both be equal
to 𝐴 squared or indeed 𝐵 squared.
We could of course confirm this by
multiplying the matrices together longhand and checking that the two products do
indeed give the same result. But there is no need. Once we spotted that the two
matrices are identical, we know that their product will be the same regardless of
which way around we find it.
So we can conclude that, for these
two particular matrices 𝐴 and 𝐵, but not in general, 𝐴𝐵 is equal to 𝐵𝐴.