 Solution

The unique symmetric and antisymmetric parts of a matrix are $S=(M+M^T)/2$ and $A=(M-M^T)/2$, givingTheir matrix products with their transposes are $S^{T}S=(a+b{)}^{2}I/4$ and $A^{T}A=(a-b{)}^{2}I/4$. Each is an orthogonal matrix precisely when the corresponding factor is one. Thus $a+b=\pm 2$ and $a-b=\pm 2$, with independent sign choices, and the simultaneous possibilities are