Project ForgeEngineering Mathematics
Linear Algebra
GATE 2019
Hard
The set of equations x+y+z=1, ax−ay+3z=5, 5x−3y+az=6 has infinite solutions, if a=
- A−3
- B3
- 4
- D−4
Explanation
Infinite solutions require the coefficient-matrix determinant to vanish. Expanding det[[1,1,1],[a,−a,3],[5,−3,a]] gives −2a²+2a+24=0, i.e. a²−a−12=0, so (a−4)(a+3)=0, giving a=4 or a=−3. Checking consistency of the augmented system confirms a=4 gives infinitely many solutions (a=−3 makes the system inconsistent instead).
Source: GATE ME 2019 Session 1 (Set 1) (2019)
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