If `A=[(alpha, 2),(2,alpha)]` and `|A^(3)|=125` then `alpha` is

Options

A. `+-1`
B. `+2`
C. `+-3`
D. `+-5`

Answer

To solve the problem, we need to find the value of \( \alpha \) given that the matrix \( A = \begin{pmatrix} \alpha & 2 \\ 2 & \alpha \end{pmatrix} \) and \( |A^3| = 125 \).

### Step-by-Step Solution:

1. **Calculate the Determinant of Matrix A**:
The determinant of a 2x2 matrix \( A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) is given by the formula:
\[ |A| = ad - bc \]
For our matrix \( A \):
\[ |A| = \alpha \cdot \alpha - 2 \cdot 2 = \alpha^2 - 4 \]

2. **Use the Property of Determinants**:
We know that for any square matrix \( A \), the determinant of \( A^n \) is given by:
\[ |A^n| = |A|^n \]
Therefore, for \( n = 3 \):
\[ |A^3| = |A|^3 \]
Given that \( |A^3| = 125 \), we can write:
\[ |A|^3 = 125 \]

3. **Solve for |A|**:
Taking the cube root of both sides, we find:
\[ |A| = 5 \]

4. **Set Up the Equation**:
From step 1, we have:
\[ \alpha^2 - 4 = 5 \]

5. **Solve for \( \alpha^2 \)**:
Rearranging the equation gives:
\[ \alpha^2 = 5 + 4 = 9 \]

6. **Find the Values of \( \alpha \)**:
Taking the square root of both sides, we get:
\[ \alpha = \pm 3 \]

### Conclusion:
The possible values of \( \alpha \) are \( 3 \) and \( -3 \).