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Given three matrices $A, B,C \in \mathbb{Z}^{n \times n}$ we want to test whether $AB \neq C$. Assume that the arithmetic operations $+$ and $-$ take constant time when applied to numbers from $\mathbb{Z}$.

How can I state an algorithm with one-sided error that runs in $O(n^2)$ time and prove its correctness?

I tried it now for several hours but I can't get it right. I think I have to use the fact that for any $x \in \mathbb{Z}^n$ at most half of the vectors $s \in S = \left\{1, 0\right\}^n$ satisfy $x \cdot s = 0$, where $x \cdot s$ denotes the scalar product$\sum_{i=1}^{n} x_is_i$.

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    $\begingroup$ In the last paragraph: you need “for any nonzero x,” which is obvious but crucial for a solution. $\endgroup$ – Tsuyoshi Ito May 13 '12 at 18:08
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If $AB=C$, then $A(Bx)=Cx$ for all vectors $x$. Generate vectors randomly and check. This known as Freivalds' algorithm. Wikipedia has details.

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  • $\begingroup$ But i Need a One-sided Error algorith. Can someone help me? $\endgroup$ – Queue May 13 '12 at 22:35
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    $\begingroup$ What makes you think this isn't one-sided? (It is.) $\endgroup$ – rgrig May 14 '12 at 8:03

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