For which values of a will the following systems of linear equations have no solutions, one solution , or an infinite number of solutions?
x+2y-3z=4
2x-y+5z=2
4x+3y+a^2z=a+3
Several functions in Mathematica can do it. Example: use the Reduce function:
Reduce[{x + 2 y - 3 z == 4 2 x - y + 5 z == 2 4 x + 3 y + a^2 z == a + 3}, {x, y, z}]
From the result you should always get the linear system solvable for a \in \mathbb{R}