Question

The eigenvalues of matrix 

are 5 and 10. For matrix \( B = A + \alpha I \), where \( \alpha \) is a constant and \( I \) is the \( 2 \times 2 \) identity matrix, its eigenvalues are 
 

• A. 5, 10
• B. \( 5 + \alpha, 10 + \alpha \)
• C. \( 5 - \alpha, 10 - \alpha \)
• D. \( 5\alpha, 10\alpha \)

Hint:

When adding a scalar multiple of the identity matrix to a matrix, the eigenvalues are shifted by that scalar.

Answer 1

The Correct Option is B

We are given that the eigenvalues of matrix 

are 5 and 10. Now, we are asked to find the eigenvalues of matrix \( B \), which is given by \( B = A + \alpha I \), where \( I \) is the identity matrix. Step 1: Eigenvalues of matrix \( A \). The eigenvalues of matrix \( A \) are given as 5 and 10. We can express this as: \[ \text{Eigenvalues of } A: \lambda_1 = 5, \lambda_2 = 10. \] Step 2: Effect of adding \( \alpha I \). When a constant multiple of the identity matrix \( \alpha I \) is added to a matrix, the eigenvalues of the matrix are shifted by the constant \( \alpha \). Therefore, the eigenvalues of matrix \( B \) will be: \[ \lambda_B = \lambda_A + \alpha. \] Thus, the eigenvalues of matrix \( B \) are: \[ \lambda_{B1} = 5 + \alpha, \quad \lambda_{B2} = 10 + \alpha. \] Therefore, the correct answer is (B). Final Answer: (B) \( 5 + \alpha, 10 + \alpha \)