Question

If \[ A = \begin{bmatrix} -7 & 3 \\ 3 & -1 \end{bmatrix} \] then \( \det(A^5) \) is equal to:

• A. \( 81 \)
• B. \( -81 \)
• C. \( 243 \)
• D. \( -243 \)
• E. \( -32 \)

Hint:

For matrix exponentiation, \( \det(A^n) = (\det A)^n \) holds for square matrices.

Answer 1

Answer: (E)

Step 1: Compute \( \det(A) \) \[ \det(A) = \begin{vmatrix} -7 & 3 \\ 3 & -1 \end{vmatrix} \] Using the determinant formula: \[ \det(A) = (-7 \times -1) - (3 \times 3) = 7 - 9 = -2 \] Step 2: Compute \( \det(A^5) \) Using the determinant property: \[ \det(A^n) = (\det A)^n \] For \( n = 5 \): \[ \det(A^5) = (-2)^5 = -32 \] Final Answer: \[ \boxed{-32} \]