[2 points] Given two matrices A and B of size = 2* 2  such that detA = –56 and detB = –8, calculate the following determinant:  det( 4A^ T *B^-1 ) 

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.

Given two matrices A and B of size = 2* 2 such that det(A) = –56 and det(B) = –8, calculate the following determinant: det( 4A^ T *B^-1 )

Learn how to calculate the determinant of 4A^T * B^-1, given that det(A) = -56 and det(B) = -8. This problem uses key linear algebra concepts such as scalar multiplication, transpose, and inverse matrix properties. Suitable for students in undergraduate linear algebra courses.

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