Involutory matrix

In mathematics, an involutory matrix is a matrix that is its own inverse. That is, multiplication by matrix A is an involution if and only if A² = I. Involutory matrices are all square roots of the identity matrix. This is simply a consequence of the fact that any nonsingular matrix multiplied by its inverse is the identity.

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

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.

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