Question Bank • QB_EM_135 NAT • 2 Marks

Section 1: Engineering Mathematics

Linear Algebra

Subtopic: Cayley–Hamilton theorem

Question Statement:

A $2 \times 2$ matrix $A$ has trace equal to $5$ and determinant equal to $6$. According to the Cayley-Hamilton theorem, $A^2 - 5A + kI = 0$. What is the value of the scalar constant $k$?

Answer Key:

6

Step-by-Step Solution & Explanation

The characteristic equation of a $2 \times 2$ matrix is: $\lambda^2 - \text{tr}(A)\lambda + \det(A) = 0 \implies \lambda^2 - 5\lambda + 6 = 0$ By the Cayley-Hamilton theorem, every square matrix satisfies its own characteristic equation: $A^2 - 5A + 6I = 0$ Thus $k = 6$.
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