# Involutory Matrix – Definition and Examples

Here you will learn what is involutory matrix with examples.

Let’s begin –

## Involutory Matrix

If $$A^2$$ = I . the matrix A is said to be an involutory matrix, i.e. the square roots of the identity matrix (I) is involutory matrix.

Note : The determinant value of this matrix (A) is 1 or -1.

Example : Show that the matrix A = $$\begin{bmatrix} -5 & -8 & 0 \\ 3 & 5 & 0 \\ 1 & 2 & -1 \end{bmatrix}$$ is involutory.

Solution : We have,

A = $$\begin{bmatrix} -5 & -8 & 0 \\ 3 & 5 & 0 \\ 1 & 2 & -1 \end{bmatrix}$$

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

$$\implies$$ $$A^2$$ = $$\begin{bmatrix} -5 & -8 & 0 \\ 3 & 5 & 0 \\ 1 & 2 & -1 \end{bmatrix}$$ $$\times$$ $$\begin{bmatrix} -5 & -8 & 0 \\ 3 & 5 & 0 \\ 1 & 2 & -1 \end{bmatrix}$$

= $$\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$$

Hence, the given matrix A is involutary.

Example : Show that the sqare matrix A is involutary, iff (I – A) (I + A) = O

Solution : Let A is the involutary matrix,

Then, $$A^2$$ = I

(I – A) (I + A) = $$I^2$$ + IA – AI – $$A^2$$

= I + A – A – $$A^2$$

= I – $$A^2$$ = O

Conversely, let (I – A) (I + A) = O

$$\implies$$ $$I^2$$ + IA – AI – $$A^2$$ = O

$$\implies$$ I + A – A – $$A^2$$ = O

= I – $$A^2$$ = O

Hence, the given matrix A is involutary.