Answer to Question #345562 in Linear Algebra for Zee

Question

Find an expression for a square matrix A satisfying A squared

= In, where In is the n × n identity matrix. Give 3

examples for the case n = 3. 

Answer 1

Matrix $A$ is an involutory matrix, if $A^2=I_n$ .

There is no formula for this type matrix, but there many examples.

All matrix $A=\begin{pmatrix} \pm 1 &0&0&…&0 \\ 0& \pm1&0&…&0 \\ 0&0&\pm1&…&0 \\ \vdots &\vdots &\vdots& \ddots &0 \\ 0&0&0&0&\pm1 \end{pmatrix}$ satisfy $A^2=I_n$ .

Examples of $3\times 3$ matrices:

$A=\begin{pmatrix} 1&0&0\\0&1&0\\0&0&1 \end{pmatrix}$

$B=\begin{pmatrix} -1&0&0\\0&1&0\\0&0&-1 \end{pmatrix}$

$C=\begin{pmatrix} 1&0&0\\0&0&-1\\0&-1&0 \end{pmatrix}$