Row Reduction and Echelon Forms

Echelon Form

The good boxes (Echelon form matrices and row reduced echelon form matrices)

Is matrix is in echelon form if:

• all nonzero rows are above any zero row
• each leading entry is in a column to the right of the leading entry of the row above it

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

A pivot is the first non-zero row in a matrix!

Row Reduced Echelon Form

1. all the requirements for echelon form
2. make all the elements in the column of a leading term equal to zero
3. all leading terms must be 1s

BACK TO OUR EQUATION

Let’s try to get this in REF:

$$\left[\begin{array}{ccc}1& 5& 6\\ 2& 7& -3\end{array}\right]$$

Try to get it to echelon forme (add $-2R_1$ to $R_2$)

$$\left[\begin{array}{ccc}1& 5& 6\\ 0& -3& -15\end{array}\right]$$

Multiply $R_1$ by 3 and $R_2$ by 5

$$\left[\begin{array}{ccc}3& 15& 18\\ 0& 7-15& -75\end{array}\right]$$

Add $R_2$ to $R_1$

$$\left[\begin{array}{ccc}3& 0& -27\\ 0& -15& -75\end{array}\right]$$

Multiply $R_1$ by $\frac{1}{3}$ and $R_2$ by $-\frac{1}{15}$

$$\left[\begin{array}{ccc}1& 0& -19\\ 0& 1& 5\end{array}\right]$$

We are now in REF.

This may be rewritten as $x+0y=-19$ and $0x+y=5$ or $x=-19$, $y=5$.