True or False Quiz About a System of Linear Equations
Problem 78
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Question 1 of 3
1. Question
True or False. A linear system of four equations in three unknowns is always inconsistent.
Correct
Good! For 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.Incorrect
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. 
Question 2 of 3
2. Question
True or False. A linear system with fewer equations than unknowns must have infinitely many solutions.
Correct
Good! For example, consider the system of one equation with two unknowns
\[0x+0y=1.\]
This system has no solution at all.Incorrect
For example, consider the system of one equation with two unknowns
\[0x+0y=1.\]
This system has no solution at all. 
Question 3 of 3
3. Question
True or False. If the system $A\mathbf{x}=\mathbf{b}$ has a unique solution, then $A$ must be a square matrix.
Correct
Good! For example, 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.Incorrect
For example, 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.
(Purdue University Linear Algebra Exam)
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