What is the largest $k$ such that if you can multiply $3 \times 3$ matrices using $k$ multiplications (not assuming commutativity of multiplication), then you can multiply $n \times n$ matrices is time $o(n^{\lg7})$? What would the running time of this algorithm be?

If we apply divide and conquer on this, we get the following recurrence:

$$ T(n) = kT(n/3) + \Theta(n) $$

The master method tells us that if $k$ is large enough, the complexity of this recurrence is $\Theta(n^{\log_{3}k})$. We need to solve:

$$ n^{\log_3k} < n^{\lg7} \\ \Downarrow \\ \log_3k < \lg7 \\ \Downarrow \\ k < 3^{\lg7} \approx 21.84986 $$