Determine the largest integer $$ such that if there is a way to multiply $3\backslash$times $3$ matrices using $$

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

algorithm to multiply $\backslash$times $$ matrices $($where $$ is an exact power of $3)$ in time $$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 $$

$3$

$\backslash$times $$

$3$

$.)$