True or False Quiz About a System of Linear Equations

Purdue University Linear Algebra Exam Problems and Solutions

Problem 78

Determine whether the following sentence is True or False.

(Purdue University Linear Algebra Exam)

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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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