Exercise 12, § 8

A matrix $(\alpha_{ij})$ of order $n$ is called centrosymmetric (resp. skew-centrosymmetric) if $\alpha_{n-i+1, n-j+1} = \alpha_{ij}$ (resp. $\alpha_{n-i+1, n-j+1} = -\alpha_{ij}$) for $1 \leq i \leq n, 1 \leq j \leq n$.

(a) Show that the determinant of a centrosymmetric matrix of even order $2p$ can be expressed in the form of a product of two determinants of order $p$ and the determinant of a centrosymmetric matrix of order $2p + 1$ in the form of a product of a determinant of order $p$ and a determinant of order $p + 1$.

(b) Show that the determinant of a skew-centrosymmetric matrix of even order $2p$ can be expressed in the form of a product of two determinants of order $p$. The determinant of a skew-centrosymmetric matrix of odd order $2p + 1$ is zero if in $A$ the relation $2\xi = 0$ implies $\xi = 0$; otherwise it can be expressed in the form of the product of $\alpha_{p+1, p+1}$ by two determinants of order $p$.