Let $A$ be a real $7\times 3$ matrix such that its null space is spanned by the vectors
\[\begin{bmatrix}
1 \\
2 \\
0
\end{bmatrix}, \begin{bmatrix}
2 \\
1 \\
0
\end{bmatrix}, \text{ and } \begin{bmatrix}
1 \\
-1 \\
0
\end{bmatrix}.\]
Then find the rank of the matrix $A$.

(Purdue University, Linear Algebra Final Exam Problem)

Determine whether the following sentence is True or False.

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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.