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if A=[{:(alpha,2),(2, alpha):}] and |A|^...

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

A

`_|_1`

B

`+-2`

C

`+-3`

D

`+-5`

Text Solution

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The correct Answer is:
To solve the problem, we need to find the value of \( \alpha \) given the matrix \( A = \begin{pmatrix} \alpha & 2 \\ 2 & \alpha \end{pmatrix} \) and the condition that \( |A|^3 = 125 \). ### Step-by-step Solution: 1. **Understanding the Determinant Condition**: We know that \( |A|^3 = 125 \). This implies that \( |A| = 5 \) because \( 5^3 = 125 \). 2. **Calculating the Determinant of Matrix A**: The determinant of matrix \( A \) can be calculated using the formula for the determinant of a 2x2 matrix: \[ |A| = \alpha \cdot \alpha - 2 \cdot 2 = \alpha^2 - 4 \] 3. **Setting Up the Equation**: Since we found that \( |A| = 5 \), we can set up the equation: \[ \alpha^2 - 4 = 5 \] 4. **Solving for \( \alpha \)**: Rearranging the equation gives: \[ \alpha^2 = 5 + 4 = 9 \] Taking the square root of both sides, we find: \[ \alpha = \pm 3 \] 5. **Final Values of \( \alpha \)**: Therefore, the possible values of \( \alpha \) are: \[ \alpha = 3 \quad \text{or} \quad \alpha = -3 \] ### Summary: The values of \( \alpha \) are \( 3 \) and \( -3 \).
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