GATE 2026 EC Q.41: solution and answer

Official answer: CSource: Official answer key, GATE 2026 EC (IIT Guwahati)

1. With no direct feed-through term ($D=0$), the transfer function of a state-space model is $$G(s)=\frac{Y(s)}{U(s)}=C(sI-A{)}^{-1}B=\frac{C\text{adj}(sI-A)B}{\det (sI-A)}$$ where $A=\left[\begin{array}{cc}-4& -1.5\\ 4& 0\end{array}\right]$, $B=\left[\begin{array}{c}2\\ 0\end{array}\right]$ and $C=\left[\begin{array}{cc}1.5& 0.625\end{array}\right]$.

2. Form $sI-A$: $$sI-A=\left[\begin{array}{cc}s+4& 1.5\\ -4& s\end{array}\right]$$

3. Characteristic polynomial: $$\det (sI-A)=s(s+4)-(1.5)(-4)=s^2+4s+6$$ The constant term is $+6$, which rules out options (A) and (B).

4. Adjugate of a $2\times 2$ matrix (swap the diagonal entries, negate the off-diagonal entries): $$\text{adj}(sI-A)=\left[\begin{array}{cc}s& -1.5\\ 4& s+4\end{array}\right]$$

5. Multiply by $B$: $$\text{adj}(sI-A)B=\left[\begin{array}{cc}s& -1.5\\ 4& s+4\end{array}\right]\left[\begin{array}{c}2\\ 0\end{array}\right]=\left[\begin{array}{c}2s\\ 8\end{array}\right]$$

6. Multiply by $C$: $$\left[\begin{array}{cc}1.5& 0.625\end{array}\right]\left[\begin{array}{c}2s\\ 8\end{array}\right]=1.5(2s)+0.625(8)=3s+5$$ The numerator has a constant term $5$, which rules out option (D).

7. Therefore $$\frac{Y(s)}{U(s)}=\frac{3s+5}{s^2+4s+6}$$ which is option (C). Quick check: the DC gain $G(0)=-C{A}^{-1}B$ should equal $\frac{5}{6}$. Here $A^{-1}=\frac{1}{6}\left[\begin{array}{cc}0& 1.5\\ -4& -4\end{array}\right]$, so $A^{-1}B=\frac{1}{6}\left[\begin{array}{c}0\\ -8\end{array}\right]$ and $-C{A}^{-1}B=-\frac{1}{6}(0.625)(-8)=\frac{5}{6}$, which agrees.