Determine the values of a for which the system has no solutions, exactly one solution, or infinitely many solutions.

x + 2y + z = 2

2x - 2y + 3z = 1

x + 2y - (a^2 - 3)z = a

I made an augmented matrix.

I tried doing that elimination thing. I ended up with (-a^2 + 2) = a - 2 for the final line.

In order for the system to be inconsistent, -(a^2 + 2) has to equate to 0, so I solved for a and got rad 2.

So it's rad 2 for no solution.

A =/= radical 2 for unique solution (I think).

I'm not sure about infinite solutions. Any hints would be greatly appreciated.