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I have a system of equations with the question "for which values of a the system has: a single solution, no solution and infinite number of solutions". In addition, I have some solution, and I need to find the mistake in the solution, I need some help with it....

So, for which values of a, the next system:

2x+y=1

4x+ay+z=0

3y+az=2

has a single solution, no solution and infinite number of solutions ?

if a=2, the matrix has proportional rows (r2 and r3), and we get infinite number of solutions. However, if we set a=2 in the original system, there is a single solution.

Where is the mistake ?

Thanks !