The basics of matrices.

A $n \times n$ non-singular square matrix $A$ is said to be involutory if the matrix is its own inverse, i.e. iff $A^2 = I_n$ . Using this definition and taking determinants of both sides, we obtain

$|A^2| = 1 \implies |A|^2 = 1 \implies |A| = \pm 1$

Hence, the determinant of an involutory matrix is always $\pm 1$ .

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Is the converse also true for all matrices?

That is if a matrix has determinant $\pm 1$ , is the matrix always an involutory matrix?