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Given that \( A \) is a \( 4 \times 4 \) matrix and \( \det(A) = -2 \), we need to find \( \det(2A^{-1}) \).

We can use the following properties of determinants:
1. \( \det(A^{-1}) = \frac{1}{\det(A)} \).
2. If \( c \) is a scalar and \( A \) is an \( n \times n \) matrix, then \( \det(cA) = c^n \cdot \det(A) \).

Applying these properties:

\[
\det(2A^{-1}) = 2^4 \cdot \det(A^{-1}) = 16 \cdot \frac{1}{\det(A)} = 16 \cdot \frac{1}{-2} = \frac{16}{-2} = -8
\]

So, \( \det(2A^{-1}) = -8 \).

Would you like more details or have any questions?

Here are some related questions:
1. What is the determinant of \( 3A^{-1} \) for a \( 3 \times 3 \) matrix \( A \)?
2. How does the determinant change if a matrix is multiplied by a scalar?
3. What is the determinant of \( (A^T)^{-1} \) where \( A^T \) is the transpose of \( A \)?
4. What is the effect on the determinant when you swap two rows of a matrix?
5. How do you calculate the determinant of a product of two matrices \( A \) and \( B \)?

**Tip:** Remember that the determinant of a matrix is only defined for square matrices and is highly sensitive to row operations and scalar multiplications.

If A is a 4x4 matrix and det(A) = -2, what is det(2A^{-1})?

Explore the step-by-step solution to find det(2A^{-1}) given that A is a 4x4 matrix with det(A) = -2. This problem delves into key concepts of matrix algebra, including the properties of determinants and scalar multiplication. Suitable for undergraduate-level students.

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