2.2 RREF

RREF

• A matrix is in RREF if

  • Each leading entry is a 1 (leading 1)

  • in a column with a leading 1, all other entries in the column are 0

• Example

  • Ex. 1 solve the variables

$$\begin{array}{r}x+y+2z=3\\ y+3z=5\\ 2z=14\end{array}$$

• then the resulting augmented matrix and RREF steps are

$$\left[\begin{array}{cccc}1& 1& 2& |3\\ 0& 1& 3& |5\\ 0& 0& 2& |14\end{array}\right]\frac{1}{2}{R}_3,R_1-2R_3,R_2-3R_3,R_1-R_2⟹\left[\begin{array}{cccc}1& 0& 0& 5\\ 0& 1& 0& -16\\ 0& 0& 1& 7\end{array}\right]$$

• then that gives us ($x=5,y=-16,z=7$)

  • Ex. 2 find RREF and rank of

$$\left[\begin{array}{ccc}1& 2& 3\\ 2& 3& 4\\ 3& 4& 5\end{array}\right]R_2-2R_1,R_3-3R_1\left[\begin{array}{ccc}1& 2& 3\\ 0& -1& -2\\ 0& -2& -4\end{array}\right]$$

• then doing the operations $R_{3}-2R_{2},-R_{2},R_{1}-2R_{2}
  • that gives us the

$$RREF\left[\begin{array}{ccc}1& 0& -1\\ 0& 1& 2\\ 0& 0& 0\end{array}\right]$$

• looking at that we can see there is 2 leading 1s $\therefore rank=2$