Given small symmetric matrix A, calculate cofactor for large matrix B made using A.

Question

$$A = \begin{bmatrix} a_1 & a_2 & a_3\\ a_2 & a_4 & a_5\\ a_3 & a_5 & a_6 \end{bmatrix}$$ $$B = \begin{bmatrix} -1 & -1 & -1\\ -1 & -1 & -1\\ -1 & -1 & -1 \end{bmatrix}$$ how to calculate cofactor of matrix $$C= \begin{bmatrix} A & B & B\\ B & A & B\\ B & B & A \end{bmatrix}$$ size of $A$ is $N \times N$, $A$ is symmetric matrix,
matrix $A$ is repeated in matrix $C$,$K$ times only on diagonal of $C$.
E.g.,$$A= \begin{bmatrix} 2 & 3 \\ 3 & 8 \\ \end{bmatrix}$$ if $k=2$ , then $$C = \begin{bmatrix} 2 & 3 & -1 & -1 \\ 3 & 8 & -1 & -1\\ -1 & -1 & 2 & 3\\ -1 & -1 & 3 & 8\\ \end{bmatrix}$$ how to calculate cofactor of $C_{11}$
I tried to solve this question using diagonal method of calculating determinants, i could find the repeating pattern but could not convert it into equation, please help me in converting into algebraic equation.

Answer 1

This is a rank-one update of $\operatorname{diag}(A-B,\,\ldots,\,A-B)$. You can use Sherman-Morrison formula and the determinant formula for rank-one update to deal with it. With patience, you should get $$ \operatorname{adj}(C) =\det(Z)^{K-2}\pmatrix{ X+Y&Y&\cdots&Y\\ Y&X+Y&\ddots&\vdots\\ \vdots&\ddots&\ddots&Y\\ Y&\cdots&Y&X+Y} $$ when $K\ge2$, where \begin{align} Z&=A-B,\\ X&=\left(\det(Z)-Ke^T\operatorname{adj}(Z)e\right)\operatorname{adj}(Z),\\ Y&=\operatorname{adj}(Z)E\operatorname{adj}(Z) \end{align} with $e=(1,\ldots,1)^T\in\mathbb R^N$ being the all-one vector and $E=ee^T$ being the $N\times N$ all-one matrix. Now take the transpose of $\operatorname{adj}(C)$ to get the cofactor matrix.