Time complexity of comparing two $N \times N$ Matrices?

Question

So each matrix has $N^{2}$ elements, and so just by comparing each element we would be doing $O(N^{2})$ operations. Is there any other way to compare these two matrices such that the number of operations is less than $O(N^{2})$ or is the matrix comparison lower bound also $\Omega(N^2)$?

Answer 1

A very simple adversary argument shows that when comparing two vectors of length $M$ (in your case, $M = N^2$), you must query (in the worst case) all positions of both vectors to know whether they are equal. That takes time $\Omega(M)$.