GATE ME 2024 Set 1 — Question 36
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Engineering Mathematics → Linear Algebra → Eigenvalues & Eigenvectors
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Question
The matrix (where ) has a negative eigenvalue if is greater than
Correct answer
(A) 3/8
Solution
Let the matrix be .
The trace of the matrix is . Since the trace is the sum of eigenvalues (), and we are looking for a negative eigenvalue, the other eigenvalue must be positive and greater than 4 in magnitude.
For a matrix, the product of eigenvalues is equal to the determinant: .
If one eigenvalue is negative and the other is positive, their product must be negative.
.
For a negative eigenvalue, we require :
.
The trace of the matrix is . Since the trace is the sum of eigenvalues (), and we are looking for a negative eigenvalue, the other eigenvalue must be positive and greater than 4 in magnitude.
For a matrix, the product of eigenvalues is equal to the determinant: .
If one eigenvalue is negative and the other is positive, their product must be negative.
.
For a negative eigenvalue, we require :
.
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