Question

Give an example of 2 × 2 matrix with non-zero entries that has no inverse.

Answer 1

If the determinant of a square matrix A is 0, then A has no inverse.

For example

$$A=\begin{pmatrix} 2 & 1 \\ -2 & -1 \end{pmatrix}$$

$$\text{det}A=\begin{vmatrix} 2 & 1 \\ -2 & -1 \end{vmatrix}=2(-1)-1(-2)=0$$

The 2x2 matrix A has no inverse.