Allied & ICAR Exam Bank • ALLIED_2616 MCQ • 2 Marks

Section 1: Engineering Mathematics

Linear Algebra

Subtopic: Cayley–Hamilton theorem

Question Statement:

If a $2 \times 2$ matrix $A$ satisfies $A^2 - 4A + 3I = 0$, then $A^{-1}$ can be expressed as:

Options:

  • (A) $\frac{1}{3}(4I - A)$
  • (B) $\frac{1}{3}(A - 4I)$
  • (C) $4I - 3A$
  • (D) $\frac{1}{4}(3I - A)$

Answer Key:

A

Step-by-Step Solution & Explanation

Multiplying $A^2 - 4A + 3I = 0$ by $A^{-1}$: $A - 4I + 3A^{-1} = 0 \implies 3A^{-1} = 4I - A \implies A^{-1} = \frac{1}{3}(4I - A)$.
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