Video: Determining Whether Matrix Multiplication Can Be Commutative Under Special Circumstances

Given the 2 Γ— 2 matrices 𝐴 = [8, βˆ’3 and 1, βˆ’2] and 𝐡 = [8, βˆ’3 and 1, βˆ’2], is 𝐴𝐡 = 𝐡𝐴?

02:25

Video Transcript

Given the two-by-two matrices 𝐴, which is equal to eight, negative three, one, negative two, and 𝐡, which is equal to eight, negative three, one, negative two, is 𝐴𝐡 equal to 𝐡𝐴?

So we’ve been given two two-by-two matrices. And we’re asked whether 𝐴𝐡 is equal to 𝐡𝐴. 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 products.

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 𝐡, we see that they do have another relationship. 𝐴 and 𝐡 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 𝐡. For this reason, it doesn’t matter in which order we multiply these two matrices together. The product 𝐴𝐡 will be equal to the product 𝐡𝐴. And in fact, they’ll both be equal to 𝐴 squared or indeed 𝐡 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 𝐡, but not in general, 𝐴𝐡 is equal to 𝐡𝐴.

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