I am struggle with this exercise in numerical analysis topic.
Let $A$ be an $n\times n$ nonsingular matrix and $A^{(k)}$ the matrix obtained in the $k$-th step of Gaussian elimination for $A$ with $A^{(0)}=A$. Let $A^{(k)}=(a^{(k)}_{r,s})$
and $a_k=\max_{r,s}|a^{k}_{r,s}|$.
Suppose that partial pivoting is used in the elimination. Show that:
a) $a_{k}\leq 2^{k}a_{0}$, $k=1,2,\ldots,n-1$ for arbitrary $A$, and
b) $a_{k}\leq 2-a_{0}$, $k=1,2,\ldots,n-1$ for tridiagonal matrices $A$.
I would appreciate any hint for right direction to start. I found this notation convenient.
Many thanks