Video: Possibility of Commutativity in Matrix Multiplication

Video Transcript

State whether the following statement is true or false. If 𝐴 and 𝐵 are both two-by-two matrices, then 𝐴𝐵 is never the same as 𝐵𝐴.

In order to prove that a statement is false, we simply need to find one example where the statement is not true. We are told that both of our matrices are two by two. And we will let the elements of matrix 𝐴 be 𝑎, 𝑏, 𝑐, 𝑑. Whilst we could let the elements of matrix 𝐵 have any values, in this case, we will let matrix 𝐵 be the identity matrix: one, zero, zero, one. We know that the identity matrix has ones on its leading diagonal and zeros everywhere else.

To calculate matrix 𝐴𝐵, we need to multiply 𝑎, 𝑏, 𝑐, 𝑑 by one, zero, zero, one. When multiplying matrices, we multiply the elements of each row in the first matrix by each column in the second matrix. 𝑎 multiplied by one is equal to 𝑎, and 𝑏 multiplied by zero is zero. Therefore, the first element in the matrix 𝐴𝐵 is 𝑎. Repeating this for the other rows and columns, we get the elements 𝑏, 𝑐, and 𝑑. Matrix 𝐴𝐵 is equal to 𝑎, 𝑏, 𝑐, 𝑑, which is equal to matrix 𝐴.

We will now repeat this method when multiplying matrix 𝐵, the identity matrix, by matrix 𝐴. Once again, this gives us the elements 𝑎, 𝑏, 𝑐, 𝑑. We have therefore found an example where the matrix 𝐴𝐵 is the same as the matrix 𝐵𝐴. This leads us to a general rule. When we multiply any matrix by the identity matrix, it is the same as multiplying the identity matrix by this matrix. In both cases, the original matrix remains the same. 𝐴𝐼 is equal to 𝐼𝐴, which is equal to the matrix 𝐴.

We can actually go one stage further when looking at the commutative property of matrices. We will now let matrix 𝐵 have the elements 𝑒, 𝑓, 𝑔, ℎ. Multiplying the matrices 𝐴𝐵 and 𝐵𝐴, we get the following two-by-two matrices. At first glance, matrix 𝐴𝐵 and 𝐵𝐴 appear to have nothing in common. However, we do notice that the top-left element contains 𝑎𝑒 or 𝑒𝑎 and the bottom-right element contains 𝑑ℎ or ℎ𝑑. The elements 𝑎, 𝑑, 𝑒, and ℎ are the elements on the leading diagonals of matrices 𝐴 and 𝐵, respectively. We can see that if all the other products were equal to zero, the two matrices would be the same.

Let’s consider what happens if 𝑏, 𝑐, 𝑓, and 𝑔 are all equal to zero. Matrices 𝐴𝐵 and 𝐵𝐴 are both equal to 𝑎𝑒, zero, zero, 𝑑ℎ. This is an example of a diagonal matrix, as all the elements apart from those on the leading diagonal are equal to zero. This leads us to another general rule of matrix multiplication. If 𝐴 and 𝐵 are both diagonal matrices, then the two matrices are commutative. 𝐴𝐵 is equal to 𝐵𝐴.