Find a function f$($n$)$ such that T$($n$)=\backslash$theta $($f$($n$))$ is the solution of the recursion. $[$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$)$

Following the divide$-$and$-$conquer paradigm, Team Y has developed an algorithm that takes two $\backslash ($ m $\backslash$times m $\backslash )$ matrices as input and returns their product using $\backslash ($ k $\backslash )$ scalar multiplications. Based on this algorithm, develop an $\backslash ($ n $\backslash$times n $\backslash )$ matrix multiplication algorithm, assuming $\backslash ($ n$=$m$^\{$l$\}\backslash ).$ Write the recurrence relation for the running time complexity of this algorithm. What is the time complexity of your algorithm?

$($b$)$ Assuming $\backslash ($ m$=48\backslash ),$ what is the largest value of $\backslash ($ k $\backslash )$ for which the resulting algorithm has a complexity of $\backslash ($ O$\backslash$left$($n$^\{2\mathrm{.}79\}\backslash$right$)\backslash )?$ Justify your answer.

$($c$)$ Team Y has developed an algorithm for matrix squaring with a time complexity of $\backslash ($ O$\backslash$left$($n$^\{2\mathrm{.}3\}\backslash$right$)\backslash ).$ Extend their results to develop an algorithm for matrix multiplication with the same time complexity of $\backslash ($ O$\backslash$left$($n$^\{2\mathrm{.}3\}\backslash$right$)\backslash ).$