Matrix Algebra: finding what values of a does system have nontrivial solutions

Question

Instructions are to: find what values of 'a' does system have nontrivial solutions

The original equations are

\begin{equation} x+2y+z=0 \end{equation} \begin{equation} -x-y+z=0 \end{equation} \begin{equation} 3x+4y+(a)z=0 \end{equation}

I tried to simplify it to

$[1 $ $ 2$ $ 1 |$ $ 0 ]$

$[0 $ $ 1 $ $2 | $ $0 ]$

$[0 $ $ 0 $ $ (4+a) |$ $ 0 ]$

but was not sure where to go from here, book didn't help too much, what should my next approach be?

Answer 1

Note that your system is given by $A\vec x=\vec 0$ where $$ A= \left[\begin{array}{rrr} 1 & 2 & 1 \\ -1 & -1 & 1 \\ 3 & 4 & a \end{array}\right] $$ Note that this system has non-trivial solutions if and only if $\det A=0$. But $$ \det A=a+1 $$ Hence the system has non-trivial solutions if and only if $a=-1$.