# Singular Matrix – Definition, Examples and Determinant

Here you will learn what is singular matrix definition with examples and also determinant of singular matrix.

Let’s begin –

## Singular Matrix

Definition : A square matrix is a singular matrix if its determinant is zero.

Otherwise, it is a non-singular matrix.

Also Read : How to Find the Determinant of Matrix

Example : Show that the matrix A = $$\begin{bmatrix} 1 & -3 & 4 \\ -5 & 2 & 2 \\ 4 & 1 & -6 \end{bmatrix}$$ is singular ?

Solution : The matrix A is singular, if

|A| = 0

$$\implies$$  |A| = $$\begin{bmatrix} 1 & -3 & 4 \\ -5 & 2 & 2 \\ 4 & 1 & -6 \end{bmatrix}$$

= 1 $$\begin{vmatrix} 2 & 2 \\ 1 & -6 \end{vmatrix}$$ – ( -3) $$\begin{vmatrix} -5 & 2 \\ 4 & -6 \end{vmatrix}$$ + 4 $$\begin{vmatrix} -5 & 2 \\ 4 & 1 \end{vmatrix}$$

= 1(-12 – 2) + 3(30 – 8) + 4(-5 – 8)

= -14 + 66 – 52

= 0

$$\implies$$ |A| = 0,

Hence, Matrix A is singular.

Example : For what value of x the matrix A = $$\begin{bmatrix} 1 & -2 & 3 \\ 1 & 2 & 1 \\ x & 2 & -3 \end{bmatrix}$$ is singular ?

Solution : The matrix A is singular, if

|A| = 0

$$\implies$$  $$\begin{vmatrix} 1 & -2 & 3 \\ 1 & 2 & 1 \\ x & 2 & -3 \end{vmatrix}$$ = 0

$$\implies$$  1 $$\begin{vmatrix} 2 & 1 \\ 2 & -3 \end{vmatrix}$$ + 2 $$\begin{vmatrix} 1 & 1 \\ x & -3 \end{vmatrix}$$ + 3 $$\begin{vmatrix} 1 & 2 \\ x & 2 \end{vmatrix}$$ = 0

$$\implies$$  (-6 – 2) + 2(-3 – x) + 3(2 – 2x) = 0

$$\implies$$  -8 – 6 – 2x + 6 – 6x = 0

$$\implies$$  -8x – 8 = 0  $$\implies$$ x = -1