If $$ \operatorname{det}(A)=4 $$ and $$ \operatorname{det}(B)=8 $$, compute the following determinants. If your answer involves a fraction enter it as one. $$ \operatorname{det}(A B)=32 $$ $$ \operatorname{det}\left(B^{-1}\right)=\frac{1}{8} $$ $$ \operatorname{det}\left(A^{T} A\right)= $$ Question Help:

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

2. The determinant of a product of matrices is the product of their determinants. Therefore:
   $$
   \operatorname{det}(AB)=\operatorname{det}(A)\operatorname{det}(B)=4 \cdot 8=32.
   $$

3. The determinant of an inverse matrix is the reciprocal of the determinant:
   $$
   \operatorname{det}\left(B^{-1}\right)=\frac{1}{\operatorname{det}(B)}=\frac{1}{8}.
   $$

4. The determinant of $$ A^T A $$ is:
   $$
   \operatorname{det}\left(A^{T} A\right)=\operatorname{det}\left(A^T\right) \operatorname{det}(A).
   $$
   Since the determinant of the transpose equals the determinant of the matrix, it follows that:
   $$
   \operatorname{det}(A^T) = \operatorname{det}(A)=4.
   $$
   Therefore:
   $$
   \operatorname{det}\left(A^{T} A\right)=4 \cdot 4=16.
   $$
$$ \operatorname{det}(A B) = 32, \quad \operatorname{det}\left(B^{-1}\right) = \frac{1}{8}, \quad \operatorname{det}\left(A^{T} A\right) = 16 $$

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. Question Help: