$\mathbf{F}=\mathbb{C}:$ $$\left(\begin{array}{ccc|c} 1 &\lambda -2&0&0\\ \lambda +2&-5&0&0\\ 0&0&1&1 \end{array}\right)$$
For which values of $\lambda$ the system has:
- Unique solution
- No solution
- Infinite amount of answers
Edit: I've reached that, but don't know if it has any significance: $$\left\{ \begin{array}{l}x_1 +(\lambda -2)x_2 = 0\\ (\lambda+2)x_1-5x_2 = 0 \end{array} \right.$$ $$-(\lambda-2)(\lambda+2)-5=0$$ $$-\lambda^2+4-5=0$$ $$\lambda=\sqrt{-1}=i$$
Edit: I now know that if $\lambda=i$ there are families of answers, and unique answers if $\lambda\not=i$. Is there a way to reach this using only elementary row operations?