Determinant of unspecified matrices Suppose A and B are $5\times 5$ matrices with $\det(A) = -1/3$ and $\det(B) = 6$, find the determinant of $ 2AB$.
Solution:
$$= \det(2AB) $$
$$= 2^5 \det(A)\det(B) $$
$$= (32)(-1/3)(6)$$
$$= -64$$

I understand how all of this works, except for where $2^5$ comes from, can anyone explain how this happens?

When you multiply a matrix by $2$, that multiplies all the rows of the matrix by $2$. The determinant is linear in each row, and the matrix $AB$ has five rows, so it multiplies the resulting determinant by $2^5$.
