If $A$ is an $n \times n$ involutory matrix, then show that $$\det (A) = (-1)^{n - \text{tr}(A) \over 2}$$
A matrix is involutory if it is its own inverse, $A^{-1} = A$. Thus, the eigenvalues of an involutory matrix are $\pm 1$, and its determinant is also $\pm 1$. Regarding the trace, for a given matrix of order $n$, its trace can be any integer from $-n$ to $n$. But, how to proceed? I was wrong on thinking trace can be any interger from $-n$ to $n$ for matrix of order $n$.