Question

Let \( A \) be a \(3 \times 3\) matrix and \( B = 2A^2 + A^{-1} - I \), where \( I \) is a \(3 \times 3\) identity matrix. If the eigenvalues of \( A \) are 1, –1 and 2, then the trace of \( B \) is ________

• A. 2
• B. 0
• C. \(\frac{15}{2}\)
• D. \(\frac{19}{2}\)

Hint:

Use eigenvalue operations to compute trace for functions of matrices efficiently—trace is sum of eigenvalues.

Answer 1

The Correct Option is D

Given:
\[ \lambda_1 = 1,\quad \lambda_2 = -1,\quad \lambda_3 = 2 \]

Eigenvalues of \( A^2 \) are \( \lambda_1^2, \lambda_2^2, \lambda_3^2 = 1, 1, 4 \)
Eigenvalues of \( A^{-1} \) are \( \frac{1}{\lambda_1}, \frac{1}{\lambda_2}, \frac{1}{\lambda_3} = 1, -1, \frac{1}{2} \)

Eigenvalues of \( B = 2A^2 + A^{-1} - I \) will be:
\[ \begin{aligned} \text{For } \lambda_1: &\quad 2(1) + 1 - 1 = 2 \\ \text{For } \lambda_2: &\quad 2(1) - 1 - 1 = 0 \\ \text{For } \lambda_3: &\quad 2(4) + \frac{1}{2} - 1 = 8 + 0.5 - 1 = 7.5 \end{aligned} \]

Sum of eigenvalues (Trace of \( B \)) = \( 2 + 0 + 7.5 = \frac{19}{2} \)

Final Answer: (4) \(\frac{19}{2}\)