What is Involutory Matrix? Examples & Properties

An involutory matrix is a square matrix which when multiplied by itself, gives the resultant matrix as identity matrix. In other words, matrix B is called involutory if B2 = I. Here I is the identity matrix having size same as of matrix B.

Involutory matrix Examples

The examples of 2 x 2 involutory matrices are


    \[\begin{bmatrix} 4 &-1 \\ 15& -4 \end{bmatrix}\]


    \[\begin{bmatrix} 1 &0 \\ 0& 1 \end{bmatrix}\]

The examples of 3 x 3 involutory matrices are


    \[\begin{bmatrix} 4 &3 &3 \\ -1& 0 &-1 \\ -4&-4 & -3 \end{bmatrix}\]


    \[\begin{bmatrix} 1 &0 &0 \\ 0& 1 &0 \\ 0&0 & 1 \end{bmatrix}\]

How to check whether a matrix is involutory matrix or not?

We can easily check whether a matrix is involutory or not. For this, check the square of a matrix is the identity matrix or not i.e. B2 = I, where B is a matrix and I is the identity matrix. If this condition is satisfied then the matrix is involutory. If the condition is not satisfied then the matrix is not involutory.

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