Help with finding value for a to make a system of equations trivial and non trivial (linear algebra)

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Hi I have a question regarding solving a system of equations.

Solve the system of equations such as this:

[1, 1,       7, -7]
[2, 3,      17, 11]
[1, 2, (a^2+1), 6a]

And find the value of A which makes the system has no solutions, non- trivial and only trivial solutions.

So I use gaussian elimination to get the matrix

[1, 1, 7, -7] [0, 1, 3, 25] [0, 0, 1, (6/(a+3))]

How exactly do I got about it? I Figured a cannot be 3 due to dividing by zero therefore the value which makes the system have no solutions, but how do I find the non-trivial and trivial values?