Matrix Algebra

Parametric Vector Form

Parametric Vector Form: 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,

$\displaystyle \left[\begin{array}{ccccccc} 1 & 0 & \alpha_1 & 0 & \beta_1 & 0 ... ...ta_3 & 0 & \gamma_3 \\ 0 & 0 & 0 & 0 & 0 & 1 & \gamma_4 \end{array}\right]$

Consider free variables (

and

in the above example) as general parameters (for example,

and

). Then basic variables (

, and

in the above example) can be solved with these parameters.

Parametric Vector Form: Step 2. 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

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

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 $\displaystyle \quad \mathbf{b} = \left[\begin{array}{ccc} 7 \\ -1 \\ -4 \end{array}\right]$