Supposed that A is a 4x4 matrix and det A = 6. Find:

a. det 2A
b. det (A*A)
c. det (A^-1)

Given that matrix \( A \) is a 4x4 matrix and \( \det(A) = 6 \), let's solve the following:

### a. \( \det(2A) \)

For any scalar \( c \) and an \( n \times n \) matrix \( A \), the determinant of \( cA \) is given by:

\[
\det(cA) = c^n \cdot \det(A)
\]

Here, \( c = 2 \) and \( n = 4 \) (since \( A \) is 4x4), so:

\[
\det(2A) = 2^4 \cdot \det(A) = 16 \cdot 6 = 96
\]

### b. \( \det(A^2) \)

For the determinant of the square of a matrix \( A \), we use the property that:

\[
\det(A^2) = \det(A \cdot A) = (\det(A))^2
\]

Thus:

\[
\det(A^2) = (\det(A))^2 = 6^2 = 36
\]

### c. \( \det(A^{-1}) \)

For the inverse of a matrix \( A \), the determinant is:

\[
\det(A^{-1}) = \frac{1}{\det(A)}
\]

Therefore:

\[
\det(A^{-1}) = \frac{1}{6}
\]

### Final answers:
- a. \( \det(2A) = 96 \)
- b. \( \det(A^2) = 36 \)
- c. \( \det(A^{-1}) = \frac{1}{6} \)

Would you like further details or clarifications?

Here are 5 related questions:
1. How does the determinant change if we multiply by a scalar other than 2?
2. What happens to the determinant if matrix \( A \) is not invertible?
3. How is the determinant of a product of two different matrices calculated?
4. What can you infer about a matrix if its determinant is zero?
5. How does the determinant relate to eigenvalues of a matrix?

**Tip**: When multiplying a matrix by a scalar, always raise the scalar to the power of the matrix's dimension to find the effect on the determinant.

Supposed that A is a 4x4 matrix and det A = 6. Find: a. det 2A b. det (A*A) c. det (A^-1)

Learn how to calculate the determinant of a 4x4 matrix when det(A) = 6. This solution covers key concepts in linear algebra, including scalar multiplication, matrix products, and inverse matrices. Detailed steps provided for finding det(2A), det(A^2), and det(A^-1). Suitable for undergraduate students studying linear algebra.

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