find $\det(\det(A)B[\det(B)A^{-1}])$

Question

$A$ and $B$ are matrices, let $A= \begin{bmatrix}2 & 0 & 3\\-1 & -2 & 1\\2 & 0 & 1\end{bmatrix}$ and $B=\begin{bmatrix}1 & -1 & 0\\1 & 0 & 1\\-1 & 1 & 1/3\end{bmatrix}$ find the next: $$\det\left(\det\left(A\right)B\left[\det\left(B\right)A^{-1}\right]\right)$$

Answer 1

Note that, for two square matrices $A$ and $B$ of order $n$, $\det(AB) = \det(A)\det(B)$. Therefore, for any given scalar value $k$, $$\det(kA) =\det(kIA) = \det(kI)\det(A)= k^n\det(A).$$

Utilizing the two properties, $$\det\left(\det(A)\;B\;\det(B)\; A^{-1}\right) = \det(A)^3\det(B)^3\det(BA^{-1})$$ $$=\det(A)^3\det(B)^3\det(B) \det(A^{-1}) = \det(A)^2\det(B)^4 = 8^2\frac{1}{3^4} = \frac{64}{81}$$

Remember that $\det(A)$ and $\det(B)$ are scalar values, hence we are able to use the property mentioned earlier.