The problem asks for $\det(A \cdot B)$, where matrices $A$ and $B$ are provided. Here's how we solve it:

Key Property:

$\det(A \cdot B) = \det(A) \cdot \det(B)$

Thus, we need to compute $\det(A)$ and $\det(B)$ first.

Step 1: Compute $\det(A)$

Matrix $A$ is: $A = \begin{pmatrix} 1 & 0 & -1 \\ 0 & 3 & 1 \\ 2 & 2 & -1 \end{pmatrix}.$ The determinant is computed as: $\det(A) = 1 \cdot \det\begin{pmatrix} 3 & 1 \\ 2 & -1 \end{pmatrix} - 0 \cdot \det\begin{pmatrix} 0 & 1 \\ 2 & -1 \end{pmatrix} + (-1) \cdot \det\begin{pmatrix} 0 & 3 \\ 2 & 2 \end{pmatrix}.$

Compute minors:

1. $\det\begin{pmatrix} 3 & 1 \\ 2 & -1 \end{pmatrix} = (3)(-1) - (1)(2) = -3 - 2 = -5$,
2. $\det\begin{pmatrix} 0 & 3 \\ 2 & 2 \end{pmatrix} = (0)(2) - (3)(2) = 0 - 6 = -6$.

Substitute back:

$\det(A) = 1 \cdot (-5) + 0 \cdot (\text{anything}) + (-1) \cdot (-6) = -5 + 6 = 1.$

Step 2: Compute $\det(B)$

Matrix $B$ is: $B = \begin{pmatrix} 3 & 0 & -1 \\ 1 & 3 & 1 \\ 2 & 0 & -1 \end{pmatrix}.$ The determinant is: $\det(B) = 3 \cdot \det\begin{pmatrix} 3 & 1 \\ 0 & -1 \end{pmatrix} - 0 \cdot \det\begin{pmatrix} 1 & 1 \\ 2 & -1 \end{pmatrix} + (-1) \cdot \det\begin{pmatrix} 1 & 3 \\ 2 & 0 \end{pmatrix}.$

Compute minors:

1. $\det\begin{pmatrix} 3 & 1 \\ 0 & -1 \end{pmatrix} = (3)(-1) - (1)(0) = -3$,
2. $\det\begin{pmatrix} 1 & 3 \\ 2 & 0 \end{pmatrix} = (1)(0) - (3)(2) = 0 - 6 = -6$.

Substitute back:

$\det(B) = 3 \cdot (-3) + 0 \cdot (\text{anything}) + (-1) \cdot (-6) = -9 + 6 = -3.$

Step 3: Compute $\det(A \cdot B)$

Using the property $\det(A \cdot B) = \det(A) \cdot \det(B)$: $\det(A \cdot B) = \det(A) \cdot \det(B) = (1) \cdot (-3) = -3.$

Final Answer:

$\boxed{-3}$

Would you like further clarification or a detailed breakdown? Here are related questions for practice:

1. How do you compute determinants of $4 \times 4$ matrices?
2. What are the properties of determinants under matrix addition?
3. How does the determinant change when a row is multiplied by a scalar?
4. How can you compute determinants using cofactor expansion along different rows?
5. How does row reduction affect the determinant of a matrix?

Tip: Always simplify the matrix using row/column operations to make determinant computation easier.