# Solutions of Systems of Linear Equations

## Consistent and Inconsistent Systems of Linear Equations

Definition

1. An $m\times n$ system of linear equations is
\begin{align*} \tag{*}
a_{1 1} x_1+a_{1 2}x_2+\cdots+a_{1 n}x_n& =b_1 \\
a_{2 1} x_1+a_{2 2}x_2+\cdots+a_{2 n}x_n& =b_2 \\
a_{3 1} x_1+a_{3 2}x_2+\cdots+a_{3 n}x_n& =b_3 \\
a_{m 1} x_1+a_{m 2}x_2+\cdots+a_{m n}x_n& =b_m,
\end{align*}
where $x_1, x_2, \dots, x_n$ are unknowns (variables) and $a_{i j}, b_k$ are numbers.
Thus an $m\times n$ system of linear equations consists of $m$ equations and $n$ unknowns $x_1, x_2, \dots, x_n$.
2. A system of linear equations is called homogeneous if the constants $b_1, b_2, \dots, b_m$ are all zero.
3. A solution of the system (*) is a sequence of numbers $s_1, s_2, \dots, s_n$ such that the substitution $x_1=s_1, x_2=s_2, \dots, x_n=s_n$ satisfies all the $m$ equations in the system (*).
4. The rank of a system of linear equation is the rank of the coefficient matrix.
Summary

1. For a given system of linear equations, there are only three possibilities for the solution set of the system: No solution (inconsistent), a unique solution, or infinitely many solutions.
2. The possibilities for the solution set of a homogeneous system is either a unique solution or infinitely many solutions.
3. If $m < n$, then an $m\times n$ system is either inconsistent or it has infinitely many solutions.
4. If $m < n$, then an $m \times n$ homogeneous system has infinitely many solutions.
5. If a consistent $m\times n$ system of linear equation has rank $r$, then the system has $n-r$ free variables.

=solution

### Problems

1. Determine all possibilities for the solution set of the system of linear equations described below.
(a) A homogeneous system of $3$ equations in $5$ unknowns.
(b) A homogeneous system of $5$ equations in $4$ unknowns.
(c) A system of $5$ equations in $4$ unknowns.
(d) A system of $2$ equations in $3$ unknowns that has $x_1=1, x_2=-5, x_3=0$ as a solution.
(e) A homogeneous system of $4$ equations in $4$ unknowns.
(f) A homogeneous system of $3$ equations in $4$ unknowns.
(g) A homogeneous system that has $x_1=3, x_2=-2, x_3=1$ as a solution.
(h) A homogeneous system of $5$ equations in $3$ unknowns and the rank of the system is $3$.
(i) A system of $3$ equations in $2$ unknowns and the rank of the system is $2$.
(j) A homogeneous system of $4$ equations in $3$ unknowns and the rank of the system is $2$.

2. Determine all possibilities for the number of solutions of each of the system of linear equations described below.
(a) A system of $5$ equations in $3$ unknowns and it has $x_1=0, x_2=-3, x_3=1$ as a solution.
(b) A homogeneous system of $5$ equations in $4$ unknowns and the rank of the system is $4$.
(The Ohio State University)

3. Determine all possibilities for the solution set of a homogeneous system of $2$ equations in $2$ unknowns that has a solution $x_1=1, x_2=5$.

4. 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*}

5. 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 values of $a$ so that the corresponding system is consistent.

6. 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 $a,b,c, d$ are scalars satisfying $a/d=b/e=c/f$.
(b) $A \mathbf{x}=\mathbf{0}$, where $A$ is a non-singular matrix.
(c) A homogeneous system of $3$ equations in $4$ unknowns.
(d) $A\mathbf{x}=\mathbf{b}$, where the row-reduced echelon form of the augmented matrix $[A|\mathbf{b}]$ looks as follows:
$\begin{bmatrix} 1 & 0 & -1 & 0 \\ 0 &1 & 2 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}.$ (The Ohio State University)