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.
Oct 31 '21 at 23:04

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 $$\Pr[ABr = Cr] = \Pr[r = 0] = 2^{-n}.$$