For what value(s) of $a$ does the system have nontrivial solutions?
\begin{align*}
&x_1+2x_2+x_3=0\\
&-x_1-x_2+x_3=0\\
& 3x_1+4x_2+ax_3=0.
\end{align*}

First note that the system is homogeneous and hence it is consistent. Thus if the system has a nontrivial solution, then it has infinitely many solutions.
This happens if and only if the system has at least one free variable. The number of free variables is $n-r$, where $n$ is the number of unknowns and $r$ is the rank of the augmented matrix.

To find the rank, we reduce the augmented matrix by elementary row operations.
\begin{align*}
\left[\begin{array}{rrr|r}
1 & 2 & 1 & 0 \\
-1 &-1 & 1 & 0 \\
3 & 4 & a & 0
\end{array} \right]
\xrightarrow[R_3-3R_1]{R_2+R_1}
\left[\begin{array}{rrr|r}
1 & 2 & 1 & 0 \\
0 & 1 & 2 & 0 \\
0 & -2 & a-3 & 0
\end{array} \right]\\
\xrightarrow{R_3+2R_2}
\left[\begin{array}{rrr|r}
1 & 2 & 1 & 0 \\
0 & 1 & 2 & 0 \\
0 & 0 & a+1 & 0
\end{array} \right].
\end{align*}
The last matrix is in row echelon form.
Thus if $a+1=0$, then the third row is a zero row, hence the rank is $2$. In this case we have $n-r=3-2=1$ free variable. Thus there are infinitely many solutions. In particular, the system has nontrivial solutions.

On the other hand, if $a+1\neq 0$, then the rank is $3$ and there is no free variables since $n-r=3-3=0$.

In summary, the system has nontrivial solutions exactly when $a=-1$.

Find Values of $a$ so that Augmented Matrix Represents a Consistent System
Suppose that the following matrix $A$ is the augmented matrix for a system of linear equations.
\[A= \left[\begin{array}{rrr|r}
1 & 2 & 3 & 4 \\
2 &-1 & -2 & a^2 \\
-1 & -7 & -11 & a
\end{array} \right],\]
where $a$ is a real number. Determine all the […]

Possibilities For the Number of Solutions for a Linear System
Determine whether the following systems of equations (or matrix equations) described below has no solution, one unique solution or infinitely many solutions and justify your answer.
(a) \[\left\{
\begin{array}{c}
ax+by=c \\
dx+ey=f,
\end{array}
\right.
\]
where $a,b,c, d$ […]

Summary: Possibilities for the Solution Set of a System of Linear Equations
In this post, we summarize theorems about the possibilities for the solution set of a system of linear equations and solve the following problems.
Determine all possibilities for the solution set of the system of linear equations described below.
(a) A homogeneous system of $3$ […]

Solve the System of Linear Equations Using the Inverse Matrix of the Coefficient Matrix
Consider the following system of linear equations
\begin{align*}
2x+3y+z&=-1\\
3x+3y+z&=1\\
2x+4y+z&=-2.
\end{align*}
(a) Find the coefficient matrix $A$ for this system.
(b) Find the inverse matrix of the coefficient matrix found in (a)
(c) Solve the system using […]

True or False Quiz About a System of Linear Equations
(Purdue University Linear Algebra Exam)
Which of the following statements are true?
(a) A linear system of four equations in three unknowns is always inconsistent.
(b) A linear system with fewer equations than unknowns must have infinitely many solutions.
(c) […]

Determine Whether Matrices are in Reduced Row Echelon Form, and Find Solutions of Systems
Determine whether the following augmented matrices are in reduced row echelon form, and calculate the solution sets of their associated systems of linear equations.
(a) $\left[\begin{array}{rrr|r} 1 & 0 & 0 & 2 \\ 0 & 1 & 0 & -3 \\ 0 & 0 & 1 & 6 \end{array} \right]$.
(b) […]