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.