# Idempotent Matrix – Definition and Example

Here you will learn what is idempotent matrix with examples.

Let’s begin –

## Idempotent Matrix

A square matrix is idempotent matrix provided $$A^2$$ = A.

For this matrix note the following :

(i) $$A^n$$ = A $$\forall$$ n $$\ge$$ 2, n $$\in$$ N.

(ii) The determinant value of this matrix is either 1 or 0.

Example : Show that the matrix A = $$\begin{bmatrix} 2 & -2 & -4 \\ -1 & 3 & 4 \\ 1 & -2 & -3 \end{bmatrix}$$ is idempotent.

Solution : We have,

A = $$\begin{bmatrix} 2 & -2 & -4 \\ -1 & 3 & 4 \\ 1 & -2 & -3 \end{bmatrix}$$

Now, $$A^2$$ = A.A

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

= $$\begin{bmatrix} 2 & -2 & -4 \\ -1 & 3 & 4 \\ 1 & -2 & -3 \end{bmatrix}$$ = A

Hence, matrix A is idempotent.

Example : Find the determinant of above matrix A = $$\begin{bmatrix} 2 & -2 & -4 \\ -1 & 3 & 4 \\ 1 & -2 & -3 \end{bmatrix}$$

Solution : We have,

A = $$\begin{bmatrix} 2 & -2 & -4 \\ -1 & 3 & 4 \\ 1 & -2 & -3 \end{bmatrix}$$

Now, | A | = $$\begin{vmatrix} 2 & -2 & -4 \\ -1 & 3 & 4 \\ 1 & -2 & -3 \end{vmatrix}$$

$$\implies$$ | A | = 2 $$\begin{vmatrix} 3 & 4 \\ -2 & -3 \end{vmatrix}$$ – (-2) $$\begin{vmatrix} -1 & 4 \\ 1 & -3 \end{vmatrix}$$ + (-4) $$\begin{vmatrix} -1 & 3 \\ 1 & -2 \end{vmatrix}$$

$$\implies$$ | A | = 2 (-9 + 8) + 2 (3 – 4) – 4 ( 2 – 3)

= 2(-1) + 2(-1) – 4(-1)

= -2 –  2 + 4 = 0

Hence, determinant of matrix A is 0.