Suppose that $A$ and $B$ are $4 \times 4$ matrices such that $$\det(A) = −2$$
and . $$\det(3A^T(B^2)^{-1})=2.$$
What must the $\det B$ equal?
I have tried to do this question by getting to
$$2 = 3^. \det(A^T) \det(B^2)^{-1},$$
but i am not too sure if that is right, like maybe the $3$ go to the $4$-th power because it is in a $4 \times 4$ matrix.
You're going well: $$ 2=3^4\det(A^T)\det((B^2)^{-1}) $$ Now recall that $\det(X^T)=\det(X)$ and $\det((X)^{-1})=(\det(X))^{-1}$, so you end up with $$ 2=3^4\det(A)(\det(B))^{-2} $$