GATE DA 2025 Set 1 — Question 13
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Linear Algebra → Eigenvalues & Eigenvectors → Eigenvalue Computation
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Question
The sum of the elements in each row of is 1. If , which one of the following statements is correct (for )?
Correct answer
(C) The equation Bx = 0 has infinitely many solutions
Solution
Given that the sum of the elements in each row of is 1. This implies that the vector is an eigenvector of corresponding to the eigenvalue , i.e., .The matrix is defined as . Let's evaluate the action of on the vector :Since , it follows that and .
Substituting these into the expression for :This shows that is a non-trivial solution to the homogeneous equation . In linear algebra, if a homogeneous system of linear equations has at least one non-trivial solution, it must have infinitely many solutions (specifically, any scalar multiple is also a solution).Therefore, the equation has infinitely many solutions.
Substituting these into the expression for :This shows that is a non-trivial solution to the homogeneous equation . In linear algebra, if a homogeneous system of linear equations has at least one non-trivial solution, it must have infinitely many solutions (specifically, any scalar multiple is also a solution).Therefore, the equation has infinitely many solutions.
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