A system of equations may have a unique solution, an infinite number of solutions, or no solution. Use matrices to find the general solution of the following system, if a solution exists.

Question

Quizplus · Accepted Answer

We can write the system of equations as an augmented matrix and reduce it to row echelon form:
$$
\begin{bmatrix}
1 & 2 & 3 & | & 1 \
2 & 4 & 6 & | & 2 \
3 & 6 & 9 & | & 3
\end{bmatrix}
ightarrow
\begin{bmatrix}
1 & 2 & 3 & | & 1 \
0 & 0 & 0 & | & 0 \
0 & 0 & 0 & | & 0
\end{bmatrix}
$$
The second and third rows of the row echelon form imply that $0 = 0$ and thus do not give any information about the solutions. The first row gives the equation $x_1 + 2x_2 + 3x_3 = 1$. Since $x_2$ and $x_3$ are free parameters, we can express the solution in vector form as:
$$
\begin{bmatrix} x_1 \ x_2 \ x_3 \end{bmatrix} = 
\begin{bmatrix} 1-2x_2-3x_3 \ x_2 \ x_3 \end{bmatrix} = 
\begin{bmatrix} 1 \ 0 \ 0 \end{bmatrix} - 
x_2 \begin{bmatrix} 2 \ 1 \ 0 \end{bmatrix} - 
x_3 \begin{bmatrix} 3 \ 0 \ 1 \end{bmatrix}
$$
Therefore, the system has infinitely many solutions and the general solution is as given in choice (C).

A System of Equations May Have a Unique Solution, an Infinite