In mathematics, an involutory matrix is a square matrix that is its own inverse. That is, multiplication by the matrix ${\displaystyle {\mathbf {A}}_{n\times n}}$ is an involution if and only if ${\displaystyle {\mathbf {A}}^{2}={\mathbf {I}},}$ where ${\displaystyle {\mathbf {I}}}$ is the ${\displaystyle n\times n}$ identity matrix. Involutory matrices are all square roots of the identity matrix. This is a consequence of the fact that any invertible matrix multiplied by its inverse is the identity.

The ${\displaystyle 2\times 2}$ real matrix ${\displaystyle {\begin{pmatrix}a&b\\c&-a\end{pmatrix}}}$ is involutory provided that ${\displaystyle a^{2}+bc=1.}$

The Pauli matrices in ⁠${\displaystyle M(2,\mathbb {C} )}$⁠ are involutory: $${\displaystyle {\begin{aligned}\sigma _{1}=\sigma _{x}&={\begin{pmatrix}0&1\\1&0\end{pmatrix}},\\\sigma _{2}=\sigma _{y}&={\begin{pmatrix}0&-i\\i&0\end{pmatrix}},\\\sigma _{3}=\sigma _{z}&={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}.\end{aligned}}}$$

One of the three classes of elementary matrix is involutory, namely the row-interchange elementary matrix. A special case of another class of elementary matrix, that which represents multiplication of a row or column by −1, is also involutory; it is in fact a trivial example of a signature matrix, all of which are involutory.

Some simple examples of involutory matrices are shown below.

$${\displaystyle {\begin{array}{cc}\mathbf {I} ={\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}};&\mathbf {I} ^{-1}={\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}}\\\\\mathbf {R} ={\begin{pmatrix}1&0&0\\0&0&1\\0&1&0\end{pmatrix}};&\mathbf {R} ^{-1}={\begin{pmatrix}1&0&0\\0&0&1\\0&1&0\end{pmatrix}}\\\\\mathbf {S} ={\begin{pmatrix}+1&0&0\\0&-1&0\\0&0&-1\end{pmatrix}};&\mathbf {S} ^{-1}={\begin{pmatrix}+1&0&0\\0&-1&0\\0&0&-1\end{pmatrix}}\\\end{array}}}$$ where

Any block-diagonal matrices constructed from involutory matrices will also be involutory, as a consequence of the linear independence of the blocks.

An involutory matrix which is also symmetric is an orthogonal matrix, and thus represents an isometry (a linear transformation which preserves Euclidean distance). Conversely every orthogonal involutory matrix is symmetric. As a special case of this, every reflection and 180° rotation matrix is involutory.