Question Bank • QB_EM_134 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 $\lambda_1 = 1$, $\lambda_2 = 3$, and $\lambda_3 = 5$. Calculate the determinant of the matrix $M^2$.

Answer Key:

225

Step-by-Step Solution & Explanation

The determinant of a matrix equals the product of its eigenvalues: $\det(M) = \lambda_1 \times \lambda_2 \times \lambda_3 = 1 \times 3 \times 5 = 15$ By properties of determinants: $\det(M^2) = (\det(M))^2 = 15^2 = 225$
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