GATE 2023 CE (CE2) – Question 37

Correct answer: (A) $[1,1,-1,1]^T$

Explanation

The null space of a $4\times4$ matrix of rank 2 has dimension $4-2=2$ (rank-nullity). The two given vectors
$$\mathbf v_1=[2,1,0,3]^T,\qquad\mathbf v_2=[1,0,1,2]^T$$
are linearly independent, so they form a basis of the null space. Any other vector in the null space must be $a\mathbf v_1+b\mathbf v_2$.

**Test the options.**
- **A, $[1,1,-1,1]^T$:** try $a=1,b=-1$: $\mathbf v_1-\mathbf v_2=[2-1,1-0,0-1,3-2]=[1,1,-1,1]^T$. ✓
- **B, $[2,0,1,2]^T$:** the second component gives $a\cdot1+b\cdot0=0$, so $a=0$; then the first gives $b=2$ but the third gives $b=1$. Not possible.
- **C, $[0,-2,1,-1]^T$:** the second component gives $a=-2$; the first gives $2a+b=0$ so $b=4$; the third gives $b=1$. Not possible.
- **D, $[3,1,1,2]^T$:** $a=1$ from the second; then $2+b=3$ gives $b=1$ but the fourth component gives $3+2=5\neq2$. Not possible.

Answer **A**.