Freivald’s Algorithm to check if a matrix is product of two

Given three matrices A, B and C, find if C is a product of A and B. 
Examples: 

Input : A = 1 1
 1 1
 B = 1 1
 1 1
 C = 2 2
 2 2
Output : Yes
C = A x B

Input : A = 1 1 1
 1 1 1
 1 1 1
 B = 1 1 1
 1 1 1
 1 1 1
 C = 3 3 3
 3 1 2
 3 3 3 
Output : No

A simple solution is to find product of A and B and then check if product is equal to C or not. A possible time complexity of this method is O(n^2.8874) using Stression's matrix multiplication. 

Freivalds' algorithm is a probabilistic randomized algorithm that works in time O(n²) with high probability. In O(kn²) time the algorithm can verify a matrix product with probability of failure less than 2^-k. Since the output is not always correct, it is a Monte Carlo randomized algorithm.

Steps : 

1. Generate an n × 1 random 0/1 vector r?.
2. Compute P? = A × (Br)? - Cr?.
3. Return true if P? = ( 0, 0, …, 0 )^T, return false otherwise.

The idea is based on the fact that if C is actually a product, then value of A × (Br)? - Cr? will always be 0. If the value is non-zero, then C can not be a product. The error condition is that the value may be 0 even when C is not a product.

Below is the implementation of the above approach:

Yes

Time Complexity: O(N ^ 2)
Auxiliary Space: O(N )