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:
The question requires us to find the value of \(\alpha\) given a \(2 \times 2\) matrix \(A\) and the determinant of its cube, \(|A^3| = 125\).


Step 2: Key Formula or Approach:
A key property of determinants is that the determinant of a matrix raised to a power is equal to the determinant of the matrix raised to that same power:
\[ |A^n| = |A|^n \] Also, the determinant of a \(2 \times 2\) matrix \(\begin{pmatrix} a & b\\ c & d \end{pmatrix}\) is given by \(ad - bc\).


Step 3: Detailed Explanation:
Given \(|A^3| = 125\), we can use the determinant property to write:
\[ |A|^3 = 125 \] Taking the cube root of both sides, we get:
\[ |A| = 5 \] Now, we calculate the determinant of matrix \(A\):
\[ |A| = (\alpha \times \alpha) - (2 \times 2) \] \[ |A| = \alpha^2 - 4 \] Equating this to the value we found:
\[ \alpha^2 - 4 = 5 \] \[ \alpha^2 = 9 \] Taking the square root of both sides gives:
\[ \alpha = \pm 3 \]

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