Find det A and det B

Question

So i was given the question

If $A$ is $3 \times 3$ matrix and $\det (2A^{-1})=-4=\det(A^3(B^{-1})^T)$, find $\det A$ and $\det B$.

I'm completely confused how to go about this. I could not find a similar example in my textbook or a theorem.

Answer 1

Using the rule $\det(AB)=\det(A)\det(B)$ for all square matrices $A,B$ of size $n=3$ we get $$ -4=\det(2A^{-1})=\det(2I_3)\det(A^{-1})=8\det(A)^{-1}. $$ Here $1=\det(I_3)=\det(AA^{-1})=\det(A)\det(A^{-1})$ again by the above rule, so $\det(A^{-1})=\det(A)^{-1}$. Multiply the above line by $\det(A)$ to obtain $-4\det(A)=8$. Then $\det(A)=-2$. A similar argument yields the second part.