# True or False Quiz About a System of Linear Equations

## Problem 78

Determine whether the following sentence is True or False.

(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) If the system $A\mathbf{x}=\mathbf{b}$ has a unique solution, then $A$ must be a square matrix.

## Solution.

All of them are false as we explain below.

### (a) True or False: A linear system of four equations in three unknowns is always inconsistent.

Consider any homogeneous system of four linear equations and three unknowns. Since a homogeneous system always has the solution $\mathbf{x}=\mathbf{0}$. Thus the statement (a) is false.

As an explicit example, the homogeneous system
$\left\{ \begin{array}{c} x+y+z=0 \\ 2x+2y+2z=0 \\ 3x+3y+3z=0 \end{array} \right.$ has the solution $(x,y,z)=(0,0,0)$. So the system is consistent.

### (b) True or False: A linear system with fewer equations than unknowns must have infinitely many solutions.

Consider the system of one equation with two unknowns
$0x+0y=1.$ This system has no solution at all. Hence the statement is false.

### (c) True or False:If the system $A\mathbf{x}=\mathbf{b}$ has a unique solution, then $A$ must be a square matrix.

Consider the matrix $A=\begin{bmatrix} 1 \\ 1 \end{bmatrix}$. Then the system
$\begin{bmatrix} 1 \\ 1 \end{bmatrix}[x]=\begin{bmatrix} 0 \\ 0 \end{bmatrix}$ has the unique solution $x=0$ but $A$ is not a square matrix.

A more theoretical argument is as follows. If vectors $\mathbf{v}_1,\dots, \mathbf{v}_k$ are linearly independent, then the system
$[\mathbf{v}_1 \dots \mathbf{v}_l]\mathbf{x}=\mathbf{0}$ has the unique solution

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