Suppose you are given n × n matrices A, B, C and you wish to check whether AB = C. You can...

Suppose you are given n × n matrices A, B, C and you wish to check whether AB = C. You can do this in steps using Strassen’s algorithm. In this question we will explore a much faster, O (n2) randomized test.

(a) Let v be an n-dimensional vector whose entries are randomly and independently chosen to be 0 or 1 (each with probablity 1/2). Prove that if M is a non-zero n × n matrix, then Pr[Mv = 0] ≤ 1/2.

(b) Show that Pr[ABv = Cv]<1/2 if AB ≠ C. Why does this give an O (n2) randomized test for checking whether AB = C?