GATE 2024 EC – Question 30
Let $\mathbb R$ and $\mathbb R^3$ denote the set of real numbers and the three dimensional vector space over it, respectively. The value of $\alpha$ for which the set of vectors
$$\{[2\ \ -3\ \ \alpha],\ [3\ \ -1\ \ 3],\ [1\ \ -5\ \ 7]\}$$
does not form a basis of $\mathbb R^3$ is ________.
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Correct answer: 5
Explanation
Three vectors fail to be a basis exactly when the determinant is zero. $\begin{vmatrix}2&-3&\alpha\\3&-1&3\\1&-5&7\end{vmatrix}=2(-7+15)+3(21-3)+\alpha(-15+1)=70-14\alpha$. This is zero for $\alpha=5$.