Answer
a.
\begin{align*}
-12
\end{align*}
b.
\begin{align*}
-375
\end{align*}
c.
\begin{align*}
4
\end{align*}
d.
\begin{align*}
-\frac{1}{3}
\end{align*}
e.
\begin{align*}
-27
\end{align*}
Work Step by Step
a.
\begin{align*}
& \quad\det{AB}
\\ & =\det A\det B
\\ & =-3\times 4
\\ & =-12
\end{align*}
b.
\begin{align*}
& \quad\det{5A}
\\ & =5^3\det A
\\ & =125\times(-3)
\\ & =-375
\end{align*}
c.
\begin{align*}
& \quad\det B^T
\\ & =\det B
\\ & =4
\end{align*}
d.
\begin{align*}
& \quad\det{A^{-1}}
\\ & =\frac{1}{\det A}
\\ & =\frac{1}{-3}
\\ & =-\frac{1}{3}
\end{align*}
e.
\begin{align*}
& \quad\det{A^{3}}
\\ & =\det{(AAA)}
\\ & =\left(
\det A
\right)^3
\\ & =(-3)^3
\\ & -27
\end{align*}
