Matrices & Determinants
Cayley-Hamilton for Finding Inverse — Coefficient Sum
nta_pyq_2023_apr
Grade None
Question:
If $A=\begin{pmatrix}1&5\\\lambda&10\end{pmatrix}$, $A^{-1}=\alpha A+\beta I$ and $\alpha+\beta=-2$, then $4\alpha^2+\beta^2+\lambda^2$ is equal to:
Step-by-Step Solution
Key Concept: By Cayley-Hamilton: $A^2-11A+(10-5\lambda)I=0\Rightarrow A^{-1}=\frac{1}{10-5\lambda}(-A+11I)$. So $\alpha=-\frac{1}{10-5\lambda}$, $\beta=\frac{11}{10-5\lambda}$.
$\lambda=3,\alpha=\frac{1}{5},\beta=-\frac{11}{5}$. $4\alpha^2+\beta^2+\lambda^2=14$.
Correct Answer: 3