GATE 2025 EE Q.13: solution and answer

Official answer: BSource: Official answer key, GATE 2025 EE (IIT Roorkee)

1. Write the augmented matrix: $$[\mathbf{A}|\mathbf{b}]=\left[\begin{array}{cccc}1& 1& 1& 1/3\\ -1& -1& -1& -1/3\\ 0& 1& -1& 0\end{array}\right]$$

2. Apply $R_2\to R_2+R_1$. Row 2 of the augmented matrix is exactly $-1$ times row 1, right-hand side included, so it becomes $\left[\begin{array}{cccc}0& 0& 0& 0\end{array}\right]$: $$\left[\begin{array}{cccc}1& 1& 1& 1/3\\ 0& 0& 0& 0\\ 0& 1& -1& 0\end{array}\right]$$

3. The two remaining coefficient rows $\left[\begin{array}{ccc}1& 1& 1\end{array}\right]$ and $\left[\begin{array}{ccc}0& 1& -1\end{array}\right]$ are not multiples of each other, so $\rho (\mathbf{A})=2$. (Check: $\det \mathbf{A}=0$ because rows 1 and 2 are proportional, so the rank must be less than 3.)

4. The zero row has a zero right-hand side, so no row reads $0=\text{nonzero}$. Hence $\rho ([\mathbf{A}|\mathbf{b}])=2$ as well.

5. Rank test (Rouché–Capelli): $\rho (\mathbf{A})=\rho ([\mathbf{A}|\mathbf{b}])=2<n=3$. The system is consistent and has $3-2=1$ free variable, so it has infinitely many solutions.

6. Explicitly, with $\mathbf{x}=(x,y,z)$: the third row gives $y-z=0$, so $y=z=t$. The first row gives $x+2t=\frac{1}{3}$, so $x=\frac{1}{3}-2t$. Every real $t$ gives a solution; for example $t=0$ gives $\left(\frac{1}{3},0,0\right)$. Check: $x+y+z=\frac{1}{3}$, $-(x+y+z)=-\frac{1}{3}$, $y-z=0$. The answer is (B).