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
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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.
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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.
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(c) […]

Determine Null Spaces of Two Matrices
Let
\[A=\begin{bmatrix}
1 & 2 & 2 \\
2 &3 &2 \\
-1 & -3 & -4
\end{bmatrix} \text{ and }
B=\begin{bmatrix}
1 & 2 & 2 \\
2 &3 &2 \\
5 & 3 & 3
\end{bmatrix}.\]
Determine the null spaces of matrices $A$ and $B$.
Proof.
The null space of the […]