Reduced Echelon Form

In this class, we will learn about the reduced echelon form of a matrix. The reduced echelon form is a matrix that has a variety of interesting properties.

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An Example

Example

We are presented with the following set of linear equations:

$$\begin{array}{rl}-2x_1-2x_2& =0\\ -x_2+x_3-3x_4& =1\\ 2x_4-x_5& =1\end{array}$$

Solution

$$\left(2-p+q,-2+p-q,p,\frac{1}{3}-\frac{1}{3}q,q\right)$$

This leads us to:

$$\begin{array}{rl}{x}_1& =-x_2\\ x_2& =-1+x_3-3x_4\\ x_4& =\frac{1}{3}+\frac{1}{3}{x}_5\end{array}$$

This doesn’t give us the full picture though, so we apply another set of transformations:

$$\begin{array}{rl}{x}_1& =2-x_3+x_5\\ x_2& =-2+x_3-x_5\\ x_3& =\frac{1}{3}-\frac{1}{3}{x}_5\end{array}$$

Example

We are presented with the following set of linear equations:

$$\begin{array}{rl}{x}_1+x_3-x_5& =2\\ x_2-x_3+x_5& =-2\\ x_4-\frac{1}{3}{x}_5& =\frac{1}{3}\end{array}$$

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Matrix Notation

Given the system:

$$\begin{array}{rl}2w-3x-y+2z& =-2\\ w+3y+z& =6\\ 2w-3x-y+3z& =-3\\ x+y+2z& =4\end{array}$$

We can make a matrix by taking the coefficients of the sets of variables, ex coefficients of w, x, y, z.

$$\left[\begin{array}{cccc}2& -3& -1& 2\\ 1& 0& 3& 1\\ 2& -3& -1& 3\\ 0& 1& 1& 2\end{array}\right]\to \left[\begin{array}{c}-2\\ 6\\ -3\\ 4\end{array}\right]$$

We can then input this into MATLAB to easily compute the result.

For almost all cases, we will be using MATLAB to solve these problems.