We are asked to compute the determinant of the matrix $\text{det}(4A^T B^{-1})$ where matrices $A$ and $B$ are both $2 \times 2$, and we are given that:

$\text{det}(A) = -56 \quad \text{and} \quad \text{det}(B) = -8$

To find $\text{det}(4A^T B^{-1})$, we can break it down step by step using the properties of determinants.

Step 1: Determinant of a scalar multiple of a matrix

For any scalar $c$ and matrix $A$, we know that: $\text{det}(cA) = c^n \text{det}(A)$ where $n$ is the size of the matrix (in this case, $n = 2$ since $A$ is $2 \times 2$). Therefore: $\text{det}(4A^T) = 4^2 \text{det}(A^T) = 16 \cdot \text{det}(A^T)$

Step 2: Determinant of the transpose

The determinant of a matrix and its transpose are the same: $\text{det}(A^T) = \text{det}(A)$ Thus: $\text{det}(4A^T) = 16 \cdot \text{det}(A) = 16 \cdot (-56) = -896$

Step 3: Determinant of the inverse

The determinant of the inverse of a matrix $B$ is: $\text{det}(B^{-1}) = \frac{1}{\text{det}(B)}$ Since $\text{det}(B) = -8$, we have: $\text{det}(B^{-1}) = \frac{1}{-8} = -\frac{1}{8}$

Step 4: Combining the results

Now, we can use the property of determinants for the product of matrices: $\text{det}(4A^T B^{-1}) = \text{det}(4A^T) \cdot \text{det}(B^{-1})$ Substitute the values we calculated: $\text{det}(4A^T B^{-1}) = (-896) \cdot \left(-\frac{1}{8}\right) = 112$

Thus, the determinant is: $\boxed{112}$

Let me know if you need further clarifications!

Related questions:

1. How do the properties of determinants simplify matrix operations?
2. What is the geometric interpretation of a determinant in $2 \times 2$ matrices?
3. How does the determinant of a matrix change under scalar multiplication?
4. How do you compute the inverse of a $2 \times 2$ matrix?
5. What are the implications if the determinant of a matrix is zero?

Tip:

Always use determinant properties, such as those involving transposes, inverses, and scalar multiplication, to simplify computations in matrix algebra.