# Proving an invertible matrix which is its own inverse has determinant $1$ or $-1$ [duplicate]

Let A be an invertible $n \times n$ matrix whose inverse is itself. Prove that $\det(A)$ is either $1$ or $-1$.

Since $A = A^{-1},\;$ we know that $A^2 = AA = I.\;$ The determinant of $\;I = 1$
$$\det(AB) = \det (A) \det (B) \implies \det(AA) = \det(I) = 1 = \det(A) \det(A)$$
What must we conclude about $\det A$? Recall, the determinant is a scalar; we may as well call $\det A = x$
$$x \times x = 1 \iff x^2 = 1 \iff x = \pm 1$$