Question Bank • QB_EM_ADV_002
NAT • 2 Marks
Section 1: Engineering Mathematics
Linear Algebra
Subtopic: Eigen values and Eigen vectors
Question Statement:
A $3 \times 3$ matrix $M$ has eigenvalues $1, 2, 3$. The trace of the matrix $(M^{-1} + M^2)$ is ________ (round off to two decimal places).
Step-by-Step Solution & Explanation
If $\lambda$ is an eigenvalue of $M$, then the corresponding eigenvalue of $(M^{-1} + M^2)$ is $\frac{1}{\lambda} + \lambda^2$.
For eigenvalues $\lambda_1 = 1, \lambda_2 = 2, \lambda_3 = 3$:
$\mu_1 = \frac{1}{1} + 1^2 = 1 + 1 = 2$
$\mu_2 = \frac{1}{2} + 2^2 = 0.5 + 4 = 4.5$
$\mu_3 = \frac{1}{3} + 3^2 = \frac{1}{3} + 9 = 9.3333$
The trace of $(M^{-1} + M^2)$ is the sum of its eigenvalues:
$\text{Trace} = 2 + 4.5 + 9.3333 = 15.8333 \approx 15.83$
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