**Noncommutativity of multiplication of matrices and existence of non-zero matrices whose product is the zero matrices (restrict to square matrices of order 2)**

Matrix multiplication is *not* commutative: **AB** ≠ **BA. **

Example: Consider the following example, calculate AB and BA

Because *A *has a dimension of 2 x 2 and *B *has a dimension of 2 x 3, the product *AB *is defined and it has dimension 2 x 3.

We can thus write:

Therefore, we define *C =AB =* [*c _{ij}*], here the entry of

*c*

_{11}is the inner product of the first row of

*A*and the first column of

*B*as shown below:

Similarly, we can calculate the remaining elements of the product as follows:

Thus, we obtain:

However, in this case the product BA is not defined. The dimensions of B and A are 2 x 3 and 2 x 2 respectively.The inner two numbers are not the same.So, the rows and columns won’t match up when we try to calculate the product.

**Example: Consider the following example, calculate AB and BA**

In this case, because both matrices *A *and *B *have dimension 2 x 2, both products *AB *and *BA *are defined, and each product is also a 2 x 2 matrix.

This shows that, in general, AB ≠ BA. In fact, here AB and BA don’t have even a single entry in common.

**Zero matrix: product of two non zero matrices**

For real number a and b, if ab = 0 then either a – 0 or b = o. But this property is not true for matrices

Example: Find AB when A = , and B =

AB =

The above example shows two non zero matrices with a zero matrix product. Hence it is not necessary that of of the matrices be a zero matrix to satisfy this property.

Example: Find two matrices A of 2X2 order such that A^{2} = 0 but A ≠ 0

Solution: Let A be a 2X2 matrix such that

Let , then

Now find a, b, c, d such that they satisfy the equality:

This means that, a + d = 0 and ad – bc = 0 (determinant=0). Hence a = -d = 1, b = -c = 1 or a = d = 0, b = 0, c = 1.

Then the matrices are not 0 but their squares are 0