$[$Rival Research Groups$]$: Two rival teams of mathematicians, Team X and Team Y$,$ are working

on similar problems. Unlike $$most researchers$,$ their primary objective is to maximize their academic

publications! To stay ahead, Team X has been carefully observing the techniques and methods

that Team Y is developing. Assist Team X in extending the techniques that Team Y has recently

developed.

$($a$)[15$ points$]$ Following the divide$-$and$-$conquer paradigm, Team Y has developed an algorithm

that takes two m$$m matrices as input and returns their product using k scalar multiplications.

Based on this algorithm, develop an n $$ n matrix multiplication algorithm, assuming n $=$ ml$.$

Write the recurrence relation for the running time complexity of this algorithm. What is the

time complexity of your algorithm?

$($b$)[5$ points$]$: Assuming m $=48,$ what is the largest value of k for which the resulting algorithm

has a complexity of O$($n$2\mathrm{.}79)?$ Justify your answer.

$($c$)[5$ points$]$: Team Y has developed an algorithm for matrix squaring with a time complexity of

O$($n$2\mathrm{.}3).$ Extend their results to develop an algorithm for matrix multiplication with the same

time complexity of O$($n$2\mathrm{.}3).$