GATE 2018 EC Q.22: solution and answer

Official answer: 2Source: Official answer key, GATE 2018 EC (IIT Guwahati)

1. A homogeneous system $\mathbf{Ax}=0$ always has the trivial solution $\mathbf{x}=0$. It has infinitely many solutions exactly when $\mathrm{rank}(\mathbf{A})<2$ (the number of unknowns). For a square matrix this is the condition $\det \mathbf{A}=0$.

2. Expand the $2\times 2$ determinant: $$\det \mathbf{A}=k\cdot k^2-2k(k^2-k)=k^3-2k^3+2k^2=-k^3+2k^2.$$

3. Factorise: $\det \mathbf{A}=k^2(2-k)$.

4. Set $\det \mathbf{A}=0$: $k^2(2-k)=0\Rightarrow k=0$ (a double root) or $k=2$. Both are real.

5. Check $k=0$: $\mathbf{A}=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$, rank $0$, so every $\mathbf{x}$ is a solution. Infinitely many solutions.

6. Check $k=2$: $\mathbf{A}=\left[\begin{array}{cc}2& 4\\ 4-2& 4\end{array}\right]=\left[\begin{array}{cc}2& 4\\ 2& 4\end{array}\right]$, rank $1$, so the system reduces to $x_1+2x_2=0$, i.e. $x_1=-2x_2$ with $x_2$ free. A one-parameter family of solutions.

7. The root $k=0$ is repeated, but it is still only one value of $k$. The distinct real values are $k=0$ and $k=2$, so the count is $2$.