If a $2x2$ matrix is not a unit or non-zero, prove that it is a zero divisor

Question

Prove that $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ $\in$ $M_2(\Bbb R)$ is a zero divisor if A is a non-zero matrix and is not a unit.

Answer 1

Since $A$ is not invertible, we have $\det A =ad-bc=0$, so $$ A\pmatrix{ d & -b\\-c& a}=0. $$

Answer 2

This works for matrices of any size $n$, viz:

If $A$ is not a unit, then $A$ is not invertible, hence

$\ker A \ne \{0\}, \tag 1$

and there exists a vector

$\vec x \ne 0 \tag 2$

such that

$A\vec x = 0; \tag 3$

let $X$ be the matrix each of the $n$ columns of which is $\vec x$:

$X = \begin{bmatrix} \vec x & \vec x & \ldots & \vec x \end{bmatrix} \ne 0; \tag 4$

then

$AX = \begin{bmatrix} A\vec x & A\vec x & \ldots & A\vec x \end{bmatrix} = 0, \tag 5$

showing that $A$ is a (left) zero divisor.