GATE 2018 EC – Question 32
Consider matrix $\mathbf A=\begin{bmatrix}k&2k\\k^2-k&k^2\end{bmatrix}$ and vector $\mathbf x=\begin{bmatrix}x_1\\x_2\end{bmatrix}$. The number of distinct real values of $k$ for which the equation $\mathbf{Ax}=\mathbf 0$ has infinitely many solutions is ________.
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Correct answer: 2
Explanation
Infinitely many solutions need $\det A=0$: $k\cdot k^2-2k(k^2-k)=-k^3+2k^2=k^2(2-k)=0$, so $k=0$ or $k=2$. That is 2 distinct values.