Let $A$ and $B$ be $3\times3$ matrices with $\det(A)=10$ and $\det(B)=12$. Find $\det(AB)$, $\det(A^4)$, $\det(2B)$, $\det((AB)^T)$.
Answers: $\det(AB)=\det(A)\det(B)=120$ , $\det(A^4)=10000$ , $\det(2B)=24$ , $\det((AB)^T)=120$
I'm really weak in this matrices, want to check my answer, any help would be appreciated, thank you.
All OK except $\det(2B)$. Remember that if you multiply a row/column of a matrix by $2$, the determinant is multiplied by $2$. If you multiply the entire matrix by $2$, then you are multiplying each row (or equivalently, each column) by $2$. Since there are $3$ rows/columns in the matrix, you are multiplying by $2$ three times, so $$\det(2B) = 2^3 \det(B) = 96.$$ It might, for your own memory, be helpful to think about a specific example. Compute the determinants of $I$ and $2I$, and verify that one is not double the other!