If $$ \operatorname{det}(A)=4 $$ and $$ \operatorname{det}(B)=8 $$, compute the following determinants. If your answer involves a fraction, enter it as one. $$ \operatorname{det}(A B)= $$ $$ \operatorname{det}\left(B^{-1}\right)= $$ $$ \operatorname{det}\left(A^{r} A\right)= $$

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We are given that  
$$
\det(A)=4 \quad \text{and} \quad \det(B)=8.
$$

1. For the matrix product $$AB$$, we have  
   $$
   \det(AB) = \det(A) \cdot \det(B) = 4 \cdot 8 = 32.
   $$

2. For the inverse of $$B$$, the formula is  
   $$
   \det(B^{-1}) = \frac{1}{\det(B)} = \frac{1}{8}.
   $$

3. For the product $$A^rA$$, note that multiplying powers of $$A$$ gives  
   $$
   A^rA = A^{r+1}.
   $$  
   Thus, using the property $$\det(A^k) = (\det(A))^k$$, we obtain  
   $$
   \det(A^{r}A) = \det(A^{r+1}) = (\det(A))^{r+1} = 4^{r+1}.
   $$

The determinants are:

$$
\det(AB)=32, \quad \det(B^{-1})=\frac{1}{8}, \quad \det(A^rA)=4^{r+1}.
$$
$$ \det(AB) = 32, \quad \det(B^{-1}) = \frac{1}{8}, \quad \det(A^rA) = 4^{r+1} $$

If and , compute the following determinants. If your answer involves a fraction,
enter it as one.

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If and , compute the following determinants. If your answer involves a fraction, enter it as one.