GATE 2023 IN – Question 63
The matrix $A=\begin{bmatrix}1&4\\-3&\alpha\\\beta&6\end{bmatrix}$ has rank one. The ratio $\alpha/\beta$ is ______ (nearest integer).
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Correct answer: -8 to -8
Explanation
Rank one requires every row to be proportional to (1,4). The second row gives $\alpha=-12$; the third gives $6=4\beta$, so $\beta=1.5$. Their ratio is $\alpha/\beta=\boxed{-8}$.