FORM GR1268 Question 24

Question:
Consider the system of linear equations
\[\begin{align}
w&+&3x&+&2y&+&2z&=0\\
w&+&4x&+&y&&&=0\\
3w&+&5x&+&10y&+&14z&=0\\
2w&+&5x&+&5y&+&6z&=0\\
\end{align}\]
with solutions of the form \((w,x,y,z)\), where \(w,x,y,\) and \(z\) are real. Which of the following statements is FALSE?

(A) The system is consistent.
(B) The system has infinitely many solutions.
(C) The sum of any two solutions is a solution.
(D) \((-5,1,1,0)\) is a solution.
(E) Every solution is a scalar multiple of \((-5,1,1,0)\).

Answer:
(E)

Answer Key:
Plug in \((-5,1,1,0)\) and confirm that it is a solution.
Thus, (D) is true.

Since the system has a solution, it is consistent.
Thus, (A) is true.

Let \(\displaystyle \mathbf A= \begin{bmatrix}1 & 3 & 2 & 2\\ 1 & 4 & 1 & 0\\ 3 & 5 & 10 & 14\\ 2 & 5 & 5& 6\end{bmatrix}\).
If \(\mathbf v=\begin{bmatrix}w\\x\\y\\z\end{bmatrix}\) is a solution,then \(\mathbf{Av}=\mathbf 0\).
Then for any scalar \(k\), we have \(k\mathbf{Av}=\mathbf{A}(k\mathbf v)=\mathbf 0\), so \(k\mathbf v\) is also a solution.
Thus, (B) is true.

If \(\mathbf v\) and \(\mathbf u\) are solutions, \(\mathbf{Av}=\mathbf 0\) and \(\mathbf{Au}=\mathbf 0\),
so \(\mathbf{Av}+\mathbf{Au}=\mathbf{A(v+u)}=\mathbf 0\).
In other words, \(\mathbf v+u\) is a solution.
Thus, (C) is true.

Since answer choices (A) through (D) are true, it must be that (E) is false.