$($This is a re$-$wording of problem $4\mathrm{.}2-4$ on page $82$ of CLRS $3^{rd}$ edition.$)$

Determine the largest integer $k$ such that if there is a way to multiply $33$ matrices using $k$

multiplications $($not assuming commutativity of multiplication of matrix elements$),$ then there is an

algorithm to multiply $nn$ matrices $($where $n$ is an exact power of $3)$ in time $o(n^{lg(7)}).$ What would

the run time of this algorithm be$?($Hint: Proceed as in the analysis of Strassen's algorithm, but recur

on submatrices of size $\frac{n}{3}\frac{n}{3})$