GATE EC 2015 Set 1 — Question 11
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Engineering Mathematics → Linear Algebra → Systems of Linear Equations
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Question
Consider a system of linear equations:The value of for which the system has infinitely many solutions is _______.
Correct answer
2 to 2
Solution
The system of equations is:
1)
2)
3) Observe that the left-hand side of equation (3) is exactly times the left-hand side of equation (1):
For the system to have infinitely many solutions, the equations must be consistent and linearly dependent. This requires the right-hand side of equation (3) to also be times the right-hand side of equation (1):
When , equation (3) becomes redundant. We are left with two independent equations in three variables, which yields infinitely many solutions. If , the system would be inconsistent (no solution).
1)
2)
3) Observe that the left-hand side of equation (3) is exactly times the left-hand side of equation (1):
For the system to have infinitely many solutions, the equations must be consistent and linearly dependent. This requires the right-hand side of equation (3) to also be times the right-hand side of equation (1):
When , equation (3) becomes redundant. We are left with two independent equations in three variables, which yields infinitely many solutions. If , the system would be inconsistent (no solution).
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