Some determinantal identities and the big cells (Q1312843)

The main results are the following theorem and a generalization of it. Given \(i_ 1,\dots,i_ k\), \(j_ 1,\dots,j_ k\), \(r_ 1,\dots,r_ k\), \(s_ 1,\dots,s_ k\), define \(M^{r_ 1\dots r_ k, s_ 1 \dots s_ k}_{i_ 1 \dots i_ k,j_ 1 \dots j_ k}\) to be the block- matrix whose \(p-q\) block, \(1 \leq p\), \(q \leq k\), is \[ a_{i_ pj_ q} \begin{pmatrix} A_{r_ ps_ q} & A_{r_ ps_{q+1}} & \cdots & A_{r_ ps_ k} \\ A_{r_{p+1}s_ q} & A_{r_{p+1}s_{q+1}} & \cdots & A_{r_{p+1}s_ k} \\ \vdots & \vdots & \vdots & \vdots \\ A_{r_ ks_ q} & A_{r_ ks_{q+1}} & \cdots & A_{r_ ksk} \end{pmatrix} \] Theorem. \(\det(M^{r_ 1\dots r_ k,s_ 1\dots s_ k}_{i_ 1\dots i_ k,j_ 1 \dots j_ k})=\prod^ k_{m=1} (i_ 1 \dots i_ m | j_ 1 \dots j_ m) (s_ m \dots s_ k | r_ m \dots r_ k)^*\) where \((i_ 1 \dots i_ m | j_ 1 \dots j_ m)\) denotes the minor consisting of rows \(i_ 1,\dots,i_ m\) and columns \(j_ 1\dots j_ m\) and \((s_ m \dots s_ k | r_ m \dots r_ k)^*\) denotes the \(k \times k\) minor formed by selection rows \(s_ m,\dots,s_ k\) and columns \(r_ m,\dots,r_ k\) from the adjoint matrix.