JEE Main 2024 (Online) 29th January Morning ShiftQuestion 1
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
Determinant
Solution
$|2A|=8|A|$ (order 3), $|A|=\alpha^2-\beta^2$. $|2A|^3=2^{21}\Rightarrow|2A|=2^7\Rightarrow8(\alpha^2-\beta^2)=128\Rightarrow\alpha^2-\beta^2=16$. $(\alpha-\beta)(\alpha+\beta)=16$ with integers of same parity: $(2,8)\to\alpha=5,\beta=3$; $(4,4)\to\alpha=4,\beta=0$. Among the options, $\alpha=5$ ($25-9=16$); $3,9,17$ give no integer $\beta$. Option B.