Let A be a CP matrix.( A completely pivoted matrix is one such that during the Gauss transformation with full pivoting there is no need to exchange rows or columns) We apply to it Gaussian elimination. I want to prove that $\begin{align} g(n,A)&=max\{1,\max_{1\leq r\leq n-1}\Big | \frac{A\begin{pmatrix} 1 &2 & \cdots &r &r+1 \end{pmatrix}} {a_{11}A\begin{pmatrix} 1 & 2 & \cdots & r \end{pmatrix}} \Big |\} \end{align}$
We consider the matrix:
\begin{align} A^{(1)}=\begin{pmatrix}a_{11} &a_{12} &\cdots &a_{1n}\\ 0 & a^{(1)}_{22} &\cdots &a^{(1)}_{2n} &\\ . & . &\cdots &.\\ . & . &\cdots &. \\ . & . &\cdots &. \\ 0 &a_{n2}^{(1)} &\cdots &a_{nn}^{(1)} \end{pmatrix} \end{align} to be the matrix after the first tranformation. Acordingly we define $A^{(r)}$
My work so far: In general $\begin{align} g(n,A)&=\frac{\max_{1<r<n-1}|a^{(r)}_{ij}|}{\max_{i,j}|a_{ij}|} \end{align}$ Since A CP $a_{11}=max_{i,j}|a_{ij}|$. Hence: $\begin{align} g(n,A)&=\frac{1}{a_{11}}\max_{i,j,k}\Big|\frac{A\begin{pmatrix} 1 &2 & \cdots &r &i \\ 1 &2 & \cdots &r &j \end{pmatrix}}{A\begin{pmatrix} 1 &2 & \cdots &r \end{pmatrix}}\Big| \\ &=\max_{1\leq r\leq n-1}\frac{1}{a_{11}\Big | A\begin{pmatrix} 1 &2 & \cdots &r \end{pmatrix}\Big|}\max_{i,j,r}\Big |A\begin{pmatrix} 1 &2 & \cdots &r &i \\ 1 &2 & \cdots &r &j \end{pmatrix}\Big | \end{align}$.
So it seems that I need to prove that $\max_{i,j,r}\Big |A\begin{pmatrix} 1 &2 & \cdots &r &i \\ 1 &2 & \cdots &r &j \end{pmatrix}\Big |=\max_{1< r \leq n-1} A\begin{pmatrix} 1 &2 &\cdots &r &r+1 \end{pmatrix}$ but I am stuck. Can someone help?