$2\mathrm{.}27.$ The square of a matrix $A$ is its product with itself, $AA.$

$($a$)$ Show that five multiplications are sufficient to compute the square of a

$22$ matrix.

$($b$)$ What is wrong with the following algorithm for computing the square of

an $nn$ matrix?

$$Use a divide$-$and$-$conquer approach as in Strassen$$s algorithm,

except that instead of getting $7$ subproblems of size $\frac{n}{2},$ we now get

$5$ subproblems of size $\frac{n}{2}$ thanks to part $($a$).$ Using the same

analysis as in Strassen's algorithm, we can conclude that the

algorithm runs in time $O(n^{lo{g}_{2}5})."$