Allied & ICAR Exam Bank • ALLIED_2614 MCQ • 1 Mark

Section 1: Engineering Mathematics

Linear Algebra

Subtopic: Cayley–Hamilton theorem

Question Statement:

For an $n \times n$ real matrix $M$ with characteristic equation $\det(M - \lambda I) = 0$, the Cayley–Hamilton theorem guarantees that substituting $M$ for $\lambda$ yields:

Options:

  • (A) The null matrix $0_{n \times n}$
  • (B) The identity matrix $I_n$
  • (C) $\text{trace}(M) I_n$
  • (D) $\det(M) I_n$

Answer Key:

A

Step-by-Step Solution & Explanation

The Cayley–Hamilton theorem asserts that if $p(\lambda) = \det(A - \lambda I) = 0$ is the characteristic polynomial of $A$, then $p(A) = 0$.
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