Question

When does a system of matrices have no solution, infinetly many solution, a unique solution.

Answer 1

I'm pretty sure it's no solution

Answer 2

The system has infinitely many solutions or has no solutions when:
$\dfrac{9}{k} = \dfrac{k}{1} =(\neq) \dfrac{9}{-3} , \\ k^2=9, \\ k=\pm 3$.

1. k=3, then $9x+3y=9 \\ 3x+y=-3$ has no solutions, because 

$\dfrac{9}{3} = \dfrac{3}{1}\neq \dfrac{9}{-3}$.

2. k=-3, then $9x-3y=9 \\ -3x+y=-3$ has infinetely many solutions, because
 $\dfrac{9}{-3} = \dfrac{-3}{1}= \dfrac{9}{-3}$.

When $k\neq \pm3$, the system has unique solution.