Matrices & Determinants
Matrix Equation — Characteristic Polynomial of C
nta_pyq_2024_apr
Grade 12
Question:
Let $B=\begin{bmatrix}1&3\\1&5\end{bmatrix}$ and $A$ be a $2\times2$ matrix such that $AB^{-1}=A^{-1}$. If $BCB^{-1}=A$ and $C^4+\alpha C^2+\beta I=O$, then $2\beta-\alpha$ is equal to:
Step-by-Step Solution
Key Concept: From $AB^{-1}=A^{-1}$: $A^2=B$, so $A=B^{1/2}$. From $BCB^{-1}=A$: $C^2=B^{-1}\cdot A\cdot B=B^{-1}AB$. Since $BCB^{-1}=A\Rightarrow BC^2B^{-1}=A^2=B$, so $C^2=B$ (similar).
$C^2=B$. Characteristic eq of $B$: $C^4-6C^2+2I=O$. $\alpha=-6$, $\beta=2$. $2\beta-\alpha=10$.
Correct Answer: 4