Proper Zero Divisor/Examples/Ring of Order 2 Complex Matrices

Example of Proper Zero Divisor
Consider the ring of square matrices $\struct {\map {\MM_\C} 2, +, \times}$ of order $2$ over the complex numbers $\C$.

We have:
 * $\begin {pmatrix} 1 & i \\ i & -1 \end {pmatrix} \begin {pmatrix} 1 & i \\ i & -1 \end {pmatrix} = \begin {pmatrix} 0 & 0 \\ 0 & 0 \end {pmatrix}$

demonstrating that $\begin {pmatrix} 1 & i \\ i & -1 \end {pmatrix}$ is a proper zero divisor of $\struct {\map {\MM_\C} 2, +, \times}$.