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.

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