Question

If \(A = \begin{pmatrix} \alpha & 2 \\ 2 & \alpha \end{pmatrix}\) and \(|A^3| = 125\), then what is the value of \(\alpha\)?

• A. \(\pm 1\)
• B. \(\pm 2\)
• C. \(\pm 3\)
• D. \(\pm 5\)

Hint:

Always use the property \(|A^n| = |A|^n\) to simplify determinant calculations of matrix powers. It prevents the need to actually multiply the matrices.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We need to determine the value of \(\alpha\) for the \(2 \times 2\) matrix \(A\), given that the determinant of \(A^3\) is 125.


Step 2: Key Formula or Approach:
We will use two fundamental determinant properties:
1. The determinant of a matrix power: \(|A^n| = |A|^n\).
2. The determinant of a \(2 \times 2\) matrix \(\begin{pmatrix} a & b \\ c & d \end{pmatrix}\) is calculated as \(ad - bc\).


Step 3: Detailed Explanation:
First, apply the power property to the given equation:
\[ |A^3| = |A|^3 = 125 \] By taking the cube root of both sides, we find:
\[ |A| = 5 \] Next, compute the determinant of matrix \(A\) directly:
\[ |A| = (\alpha)(\alpha) - (2)(2) = \alpha^2 - 4 \] Set this expression equal to the value obtained earlier:
\[ \alpha^2 - 4 = 5 \] \[ \alpha^2 = 9 \] Solving for \(\alpha\) by taking the square root yields:
\[ \alpha = \pm 3 \]

Step 4: Final Answer:
The correct choice is (C).