Zero Divisor of Ring/Examples/Order 2 Square Matrices/Example 1

Examples of Zero Divisors of Rings
Let $R$ be the ring square matrices of order $2$ over a field with unity $1$ and zero $0$.

Let:

Then:
 * $\mathbf A \mathbf B = \begin {bmatrix} 0 & 0 \\ 0 & 0 \end {bmatrix} = \mathbf B \mathbf A$

Thus both $\mathbf A$ and $\mathbf B$ are zero divisors of $R$.