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GATE 2023 AE – Question 31

Engineering Mathematics · Linear Algebra: Vector algebra, matrix algebra, systems of linear equations, rank, eigenvalues and eigenvectors · 1 mark · Numerical answer

The system $x-2y+\alpha z=0$, $2x+y-4z=0$, $x-y+z=0$ has a nontrivial solution for $\alpha=$______ (integer).

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Correct answer: 2

Explanation

Nonzero solutions require determinant zero. For coefficients $(1,-2,\alpha),(2,1,-4),(1,-1,1)$, the determinant is $6-3\alpha$, giving $\alpha=2$.