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GATE 2024 CH – Question 30

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

Consider a matrix $\mathbf{A} = \begin{bmatrix} -5 & a \\ -2 & -2 \end{bmatrix}$, where $a$ is a constant. If the eigenvalues of $\mathbf{A}$ are −1 and −6, then the value of $a$, rounded off to the nearest integer, is ____________

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Correct answer: -2

Explanation

The determinant equals the product of the eigenvalues: $(-5)(-2) - a(-2) = 10 + 2a = (-1)(-6) = 6$, so $a = -2$. (The trace $-7$ equals the sum of the eigenvalues, as it should.)