Suppose

(1)

and we would like to compute when possible.

Let’s begin with the first one

(2)

Next, we know multiplying two matrices

(3)

results in a matrix of size by . So, multiplyin two by matrices will result in a by matrix. We show the details of computation

(4)

Next

(5)

This observation shows that in general for two matrices and the rule may not be true. Next,

(6)

As you see, the result of multiplying a by matrix by a by matrix on the right is a matrix of size by However, does not exist since we cannot multiply a by matrix by a by matrix on the right. Remember that the number of columns of the first matrix must be equal to the number of rows of the second one. Next

(7)

Next, we compute

(8)

Next, we compute

(9)

As you see, in the world of matrices . This is also due to the fact that .