How do I find a value for a such that this linear system of equations is inconsistent?

Question

I have tried solving the augmented matrix but I haven't been able to get any answers from that. Can someone explain how to do a question like this? Find a value of a such that this system is inconsistent

Thank you!

Answer 1

Basically, you have to find $a$ such that no $(x, y, z)$ could possibly work.

If you sum rows 1 and 3, and compare the result to row 2, you obtain potentially mutually exclusive conditions, ie:

a) $3x + 5y + 4z = a$

and

b) $3x + 5y + (3 + a^2)z = 0$

which implies

b - a) $(3 + a^2 - 4)z = -a$

Expressed this way, the problem is simpler: you have to find $a$ such that no $z$ could possibly work.

$a = -1$ and $a = 1$ respectively lead to $0z = 1$ and $0z = -1$, both of which are unsolvable conditions.