Question Bank • QB_SUB_EM_0014 NAT • 2 Marks

Section 1: Engineering Mathematics

Linear Algebra

Subtopic: Cayley–Hamilton theorem

Question Statement:

A $2 \times 2$ matrix $A$ has trace $\text{tr}(A) = 7$ and determinant $\det(A) = 10$. By the Cayley-Hamilton theorem, $A^2 - 7A + c I = 0$. The value of scalar $c$ is:

Answer Key:

10.0

Step-by-Step Solution & Explanation

The characteristic equation of a $2 \times 2$ matrix is $\lambda^2 - \text{tr}(A)\lambda + \det(A) = 0$. By Cayley-Hamilton, $A$ satisfies its own characteristic equation: $A^2 - 7A + 10I = 0$. Thus $c = 10$.
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