Question Bank • QB_EM_ADV_019 MCQ • 1 Mark

Section 1: Engineering Mathematics

Linear Algebra

Subtopic: Cayley–Hamilton theorem

Question Statement:

A $3 \times 3$ invertible matrix $A$ has the characteristic equation $\lambda^3 - 6\lambda^2 + 11\lambda - 6 = 0$. The inverse matrix $A^{-1}$ is expressed as:

Options:

  • (A) $\frac{1}{6}(A^2 - 6A + 11I)$
  • (B) $\frac{1}{6}(-A^2 + 6A - 11I)$
  • (C) $\frac{1}{11}(A^2 - 6A + 6I)$
  • (D) $6(A^2 - 6A + 11I)$

Answer Key:

A

Step-by-Step Solution & Explanation

By the Cayley–Hamilton theorem: $A^3 - 6A^2 + 11A - 6I = O$ Isolating the identity matrix: $6I = A^3 - 6A^2 + 11A$ Multiplying both sides by $A^{-1}$: $6A^{-1} = A^2 - 6A + 11I \implies A^{-1} = \frac{1}{6}(A^2 - 6A + 11I)$
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