# Matrix Multiplication Verification

This question relates to the 2nd edition of Probability and Computation by Mitzenmacher and Upfal.

The authors then use the Law of Total Probability

My question concerns the line that begins with $$\leq$$. Shouldn't that be replaced with $$=$$ because the authors proved that the event $$\mathbf{AB} \bar{r} = \mathbf{C}\bar{r}$$ is equivalent to the event $$r_{1} = - \frac{\sum_{j=2}^n d_{1j} r_{j}}{d_{11}}$$ ?

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– D.W.
Commented Oct 31, 2021 at 23:04
• The authors do not say that $AB\bar{r} = C\bar{r}$ is equivalent to the event $r_1 = -\sum_j d_{1j}r_j / d_{11}$, rather that it implies it. Commented Jul 29, 2022 at 13:01

I'm assuming that all matrices in question are $$n \times n$$.
Suppose that $$A=B=0$$ are zero matrices and $$C=I$$ is the identity matrix. Then $$\Pr[ABr = Cr] = \Pr[r = 0] = 2^{-n}.$$