GATE CS 2017 Set 1 — Question 3
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Engineering Mathematics → Linear Algebra → System of Linear Equations
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Question
Let be scalars, not all zero, such that where are column vectors in .
Consider the set of linear equationswhere and . The set of equations has
Consider the set of linear equationswhere and . The set of equations has
Correct answer
(C) infinitely many solutions
Solution
1.Linear Dependence: We are given with not all zero. This implies that the columns of matrix are linearly dependent. Therefore, the matrix is singular (determinant is 0), and the homogeneous system has non-trivial solutions (infinitely many).
2.Consistency: The vector is given as . This can be written as a linear combination of the columns of with coefficients all equal to 1. Specifically, if we let (denoted as ), then . Thus, the system is consistent and is a particular solution.
3.Conclusion: Since the system is consistent and the matrix is singular (rank ), the general solution is given by , where is any solution to the homogeneous equation . Since there are infinitely many such , there are infinitely many solutions to .
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