General equations

STEP 1: Construct the augmented matrix  $ [\:A \hspace{0.075in} \mathbf{b}\:]$ for the matrix equation  $ A \mathbf{x} = \mathbf{b}$, then produce a reduced echelon form (REF). For example,

$\displaystyle \left[\begin{array}{ccccccc}
1 & 0 & \alpha_1 & 0 & \beta_1 & 0 &...
...\beta_3 & 0 & \gamma_3 \\
0 & 0 & 0 & 0 & 0 & 1 & \gamma_4
\end{array}\right]
$

STEP 2: Consider free variables ($ x_3$ and $ x_5$ in the above example) as general parameters (for example, $ x_3 = s$ and $ x_5 = t$). Then basic variables ($ x_1$, $ x_2$, $ x_4$, and $ x_6$ in the above example) can be solved with these parameters.

STEP 3: Together with free variables, express general solutions to  $ {A \mathbf{x} = \mathbf{b}}$ in terms of parametric vector form. In the above example, we obtain

$\displaystyle \left[\begin{array}{c}
x_1 \\
x_2 \\
x_3 \\
x_4 \\
x_5 \\
x_...
...ay}{c}
-\beta_1 \\
-\beta_2 \\
0 \\
-\beta_3 \\
1 \\
0
\end{array}\right]
$

EXAMPLE 2. Describe all solutions of $ A \mathbf{x} = \mathbf{b}$ with

$\displaystyle A =
\left[\begin{array}{ccc}
3 & 5 & -4 \\
-3 &-2 & 4 \\
6 & 1 & -8
\end{array}\right]$    and $\displaystyle \quad
\mathbf{b} =
\left[\begin{array}{ccc}
7 \\
-1 \\
-4
\end{array}\right]
$



Department of Mathematics
Last modified: 2005-10-21