Matrices & Determinants
Determinant Condition — Integer Parameters
nta_pyq_2024_jan
Grade 12
Question:
Let $A=\begin{bmatrix}1&0&0\\0&\alpha&\beta\\0&\beta&\alpha\end{bmatrix}$ and $|2A|^3=2^{21}$ where $\alpha,\beta\in\mathbb{Z}$. Then a value of $\alpha$ is
Step-by-Step Solution
Key Concept: $|A|=\alpha^2-\beta^2$. $|2A|=2^3|A|\Rightarrow|2A|^3=2^9|A|^3=2^{21}\Rightarrow|A|^3=2^{12}\Rightarrow|A|=2^4=16$. So $\alpha^2-\beta^2=16\Rightarrow(\alpha+\beta)(\alpha-\beta)=16$.
$|A|=\alpha^2-\beta^2=16$. $(\alpha+\beta)(\alpha-\beta)=16\Rightarrow\alpha=5,\beta=3$ (or $\alpha=4,\beta=0$). Answer: $\alpha=5$.
Correct Answer: 2