#11. Checking Matrix Multiplication

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Prompts for discussion:

1. Can we extend this method to distinguish between situations when $C$ is “very close” to $A \times B$ vs when $C$ is “far” from $A \times B$, for a suitable notion of distance between matrices?

2. Let’s say the Matrix Wizard always outputs $I_n$ and you are always multiplying two permutation matrices. Is it possible to count, in terms of some aspect of the permutation represented by $A \times B$, the number of vectors $\mathbf{x}$ for which $D \mathbf{x} \neq 0$?