Finding determinant of a matrix in an equation

Question

Be A and B two matrix 3x3 and

$C=(-2A)^{3}\cdot (B^{2})^{-1}$

If det(A) = -2 and det(B) = 3

find det(C)

So, don't know how to approach this problem, how do I find a matrix knowing the determinant?

Answer 1

Let's follow the hints given in the comments to use $\det(AB) = \det(A)\det(B)$ and $\det(rA) = r^n\det(A)$ (if $A$ is $n \times n$). Also recall that $\det(A^{-1}) =\det(A)^{-1}$. Then, \begin{align*}\det(C) &= \det((-2A)^3(B^2)^{-1})\\ &= \det((-2A)^3)\det((B^2)^{-1})\\ &= \det(-8A^3)\det(B^2)^{-1}\\ &=(-8)^3 \det(A)^3\det(B)^{-2}\\ &= (-8)^3 \cdot (-2^3) \cdot 3^{-2}\\ &= 4096/9 \end{align*}