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

Options

A. `pm1`
B. `pm2`
C. `pm3`
D. `pm5`

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. **Use the property of determinants**:
We know that the determinant of the cube of a matrix can be expressed as the cube of the determinant of the matrix itself. Thus, we have:
\[ |A^3| = |A|^3 \]
Given that \( |A^3| = 125 \), we can write:
\[ |A|^3 = 125 \]

2. **Express 125 as a power**:
We can express 125 as \( 5^3 \):
\[ |A|^3 = 5^3 \]
Therefore, we can conclude:
\[ |A| = 5 \]

3. **Calculate the determinant of matrix \( A \)**:
The determinant of matrix \( A \) is calculated as follows:
\[ |A| = \alpha \cdot \alpha - 2 \cdot 2 = \alpha^2 - 4 \]

4. **Set up the equation**:
Now, we equate the determinant we found with the value we derived from the previous steps:
\[ \alpha^2 - 4 = 5 \]

5. **Solve for \( \alpha \)**:
Rearranging the equation gives:
\[ \alpha^2 = 5 + 4 = 9 \]
Taking the square root of both sides, we find:
\[ \alpha = \pm 3 \]

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

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