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GATE 2024 AE – Question 51

Engineering Mathematics · Linear Algebra: Vector algebra, matrix algebra, systems of linear equations, rank, eigenvalues and eigenvectors · 2 marks · Numerical answer

Consider the matrix $A = \begin{bmatrix} 5 & k \\ -4 & -1 \end{bmatrix}$, where k is a constant. If the determinant of A is 3, then the ratio of the largest eigenvalue of A to the constant k is ___________ (rounded off to 1 decimal place).

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Correct answer: 1.49 to 1.51

Explanation

$\det A = -5 + 4k = 3$, so $k = 2$. The trace is 4 and the determinant is 3, so the eigenvalues satisfy $\lambda^2 - 4\lambda + 3 = 0$: $\lambda = 1$ and $3$. The ratio is $\frac{3}{2} = 1.5$.