$S$ is collection of $ 3 \times 3 $ Matrices with entries $0$ and $1$ real numbers. It is to be proved that number of Matrices in $S$ having determinant exactly $1$ is equal to the number of Matrices having determinant $-1$. How the bijection from these subsets of $S$ can be defined? Precisely a bijection from $S\_1$ and $S\_2$ is to be defined where $S\_1$ is the set of Matrices in $S$ having determinant exactly $1$ and $S\_2$ is the set of Matrices in $S$ having determinant exactly $-1$. Would mapping diagonal elements to anti diagonal elements work? This works for identity matrix.