Recursive relation to represent the last element of a matrix using determinant [closed]

Question

$J$ is a $N\times N$ matrix, each element of $J$ is sampled from a Gaussian distribution with zero mean and variance $N^{-1}$. The resolvent matrix is defined as $R^{(N)} = [\mathcal{E} \mathbb{I} - J]^{-1} $ where $\mathcal{E}$ is a real number. The superscript $N$ means it is a $N \times N$ matrix.The elements of $R^{(N)}$ is denoted as $R^{(N)}_{i,j}$

Another matrix is defined as $M^{(N)} = \mathcal{E} \mathbb{I} - J$. Then $$\DeclareMathOperator{\det}{det} R^{(N+1)}_{N+1,N+1} = \frac{\det[M^{(N)}]}{\det[M^{(N+1)}]} $$ I do not understand why this relation holds.
This is taken from these notes on replica theory (equation3 on page 2).

Answer 1

Check out https://en.wikipedia.org/wiki/Minor_(linear_algebra)

Cramer's rule says that $$R^{(N+1)}=\frac{1}{\det M^{(N+1)}}C^\top,$$ with $C$ the cofactor matrix of $M^{(N+1)}$. The $(N+1,N+1)$ element of $C$ is just $\det M^{(N)}$, hence $$R^{(N+1)}_{N+1,N+1}=\frac{\det M^{(N)}}{\det M^{(N+1)}}.$$