Arithmetic-geometric mean determinantal identity (Q636255)

For any \(n\) by \(n\) matrix \(A\), let \(A_{r}\left[ i,j\right] \) denote an \(r\) by \(r\) submatrix consisting of r contiguous rows and columns of \(A\), starting with row \(i\) and column \(j\). Let also the superscript \(t\) stands for transposition of a matrix and \(J_{n}\) be an all-one matrix of order \(n\). Then the theorem proves that, if \(A\) is a matrix and \(A+A^{t}=aJ_{n}\), where \( a\) is a real number, then we have the following determinantal identity \[ \underset{\text{Geometric Mean}}{\underbrace{\sqrt{\det A_{n-1}\left[ 1,1 \right] \det A_{n-1}\left[ 2,2\right] }}}=\underset{\text{Arithmetic Mean}}{ \underbrace{\frac{\det A_{n-1}\left[ 1,2\right] +\det A_{n-1}\left[ 2,1 \right] }{2}}} \]