GATE 2022 ME (ME2) – Question 46
A is a $3\times5$ 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?
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Correct answer: (B) The given set of equations will be satisfied by a zero vector of appropriate size.; (C) The given set of equations will have infinitely many solutions.
Explanation
The matrix is $3\times5$ with rank 2, so there are 5 unknowns.
**Rank-nullity theorem:**
$$\text{nullity}=\text{number of unknowns}-\text{rank}=5-2=3 .$$
So the solution space of $A\mathbf x=\mathbf 0$ has dimension 3.
- **A, unique solution:** false, because the nullity is 3, not 0.
- **B, satisfied by the zero vector:** true. A homogeneous system is always solved by $\mathbf x=\mathbf 0$.
- **C, infinitely many solutions:** true. A solution space of dimension 3 over the real numbers has infinitely many vectors (any scalar multiple of a solution is a solution).
- **D, many but finitely many:** false, for the same reason.
Correct: **B and C**.