Project ForgeA is a 3×5 real matrix of rank 2. For the set of homogeneous equations Ax=0, where 0 is a zero vector and x is a vector of unknown variables, which of the following is/are true?
- AThe given set of equations will have a unique solution.
- The given set of equations will be satisfied by a zero vector of appropriate size.
- The given set of equations will have infinitely many solutions.
- DThe given set of equations will have many but a finite number of solutions.
Explanation
By the rank-nullity theorem, the null space of A has dimension 5−rank(A) = 5−2 = 3 > 0. (A) FALSE: a unique solution requires nullity 0. (B) TRUE: x=0 always satisfies any homogeneous system, regardless of rank. (C) TRUE: a nullity of 3 means the solution set is a 3-dimensional subspace, containing infinitely many vectors. (D) FALSE: a linear homogeneous system's solution set is either just {0} or infinite — never finite-but-more-than-one.
Source: GATE ME 2022 Session 2 (ME-2) (2022)
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