Complete Solution to Ax = b

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Finding complete solution

Reduce augmented matrix to Reduced Row Echelon Form
Find particular solution
Find special (nullspace) solutions in terms of free variables
Combine particular solution with nullspace solutions

Example

\[ \begin{bmatrix} 1 & 1 & 3\\ 2 & 1 & 1\\ 3 & 2 & 4\\ \end{bmatrix} \begin{bmatrix} x\\ y\\ z\\ \end{bmatrix} = \begin{bmatrix} 1\\ 1\\ 2\\ \end{bmatrix} \]

1) Reduce augmented matrix to Reduced Row Echelon Form

\begin{bmatrix} 1 & 1 & 3 & : & 1\\ 2 & 1 & 1 & : & 1\\ 3 & 2 & 4 & : & 2\\ \end{bmatrix}

-2R1 + R2, -3R1 + R3, -1R2, 2R2 + R3, -R2 + R1

\begin{bmatrix} 1 & 0 & -2 & : & 0\\ 0 & 1 & 5 & : & 1\\ 0 & 0 & 0 & : & 0\\ \end{bmatrix}

The pivot variables are x and y and the free variable is z. To be a pivot variable it's column position on the right-hand side (so x has position 1) should have a 1 in the left-hand side column & row position (x has col 1 row 1) and anything else in that column must be a zero.

2) Find particular solution

Let the free variable = 0. Thus 1x + 0y + -2*(0) = 0, so x = 0. And 0x + 1y +5*(0) = 1. So y = 1. \[ x_{p} = \begin{bmatrix} 0\\ 1\\ 0\\ \end{bmatrix} \]

3) Find special (nullspace) solutions in terms of free variables

Use augmented matrix with the right-hand side = 0 to calculate the nullspace N(A): \[ \begin{bmatrix} 1 & 0 & -2 & : & 0\\ 0 & 1 & 5 & : & 0\\ 0 & 0 & 0 & : & 0\\ \end{bmatrix} \] Solve for x,y in terms of z.

x - 2z = 0

y + 5z = 0

x = 2z

y = -5z

\[ s_{1} = z \begin{bmatrix} 2\\ -5\\ 1\\ \end{bmatrix} \]

4) Combine particular solution with nullspace solutions

Complete solution to Ax = b is x_p + x_n.

x_n = s₁ + … s_n.

So the complete solution for this example is:

\[ \begin{bmatrix} 0\\ 1\\ 0\\ \end{bmatrix} + z \begin{bmatrix} 2\\ -5\\ 1\\ \end{bmatrix} \]