Question Bank • QB_EM_LA_007
MCQ • 2 Marks
Section 1: Engineering Mathematics
Linear Algebra
Subtopic: Cayley–Hamilton theorem
Question Statement:
Given the matrix $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$. Using the Cayley–Hamilton theorem, the matrix expression $A^2 - 5A$ is equal to:
Options:
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(A) $-2I$
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(B) $2I$
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(C) $5I$
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(D) $-5I$
Step-by-Step Solution & Explanation
The characteristic equation of $A$ is:
$\det(A - \lambda I) = (1 - \lambda)(4 - \lambda) - 6 = \lambda^2 - 5\lambda - 2 = 0$
By the Cayley–Hamilton theorem, every square matrix satisfies its own characteristic equation:
$A^2 - 5A - 2I = 0$
Rearranging terms:
$A^2 - 5A = 2I$
Therefore, option B is correct.
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