The square of a matrix A is its product with itself, AA.(a) Show that five multiplications...

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 2 × 2 matrix.

(b) What is wrong with the following algorithm for computing the square of an n × n matrix?

“Use a divide-and-conquer approach as in Strassen’s algorithm, except that instead of getting 7 subproblems of size n/2, we now get 5 subproblems of size n/2 thanks to part (a). Using the same analysis as in Strassen’s algorithm, we can conclude that the algorithm runs in time

(c) In fact, squaring matrices is no easier than matrix multiplication. Show that if n × n matrices can be squared in time O (nc), then any two n × n matrices can be multiplied in time O (nc).