Question Bank • QB_EM_ADV_022 MCQ • 1 Mark

Section 1: Engineering Mathematics

Linear Algebra

Subtopic: Eigen values and Eigen vectors

Question Statement:

A $3 \times 3$ matrix $A$ has eigenvalues $\lambda_1 = 1$, $\lambda_2 = 2$, and $\lambda_3 = -3$. The determinant of the matrix $A^T A$ is:

Options:

  • (A) $0$
  • (B) $-6$
  • (C) $36$
  • (D) $14$

Answer Key:

C

Step-by-Step Solution & Explanation

The determinant of a matrix equals the product of its eigenvalues: $\det(A) = \lambda_1 \cdot \lambda_2 \cdot \lambda_3 = (1)(2)(-3) = -6$ Using properties of determinants: $\det(A^T A) = \det(A^T) \cdot \det(A) = (\det(A))^2 = (-6)^2 = 36$
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