Find Reduced Row Echelon Form [[1,1,1,25],[7000,14000,28000,406000]]

$[\begin{array}{c} 1 & 1 & 1 & 25 \\ 7000 & 14000 & 28000 & 406000 \end{array}]$

Step 1

Perform the row operation $R_{2} = R_{2} - 7000 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & 1 & 1 & 25 \\ 7000 - 7000 \cdot 1 & 14000 - 7000 \cdot 1 & 28000 - 7000 \cdot 1 & 406000 - 7000 \cdot 25 \end{array}]$

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & 1 & 1 & 25 \\ 0 & 7000 & 21000 & 231000 \end{array}]$

Step 2

Multiply each element of $R_{2}$ by $\frac{1}{7000}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & 1 & 1 & 25 \\ \frac{0}{7000} & \frac{7000}{7000} & \frac{21000}{7000} & \frac{231000}{7000} \end{array}]$

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & 1 & 1 & 25 \\ 0 & 1 & 3 & 33 \end{array}]$

Step 3

Perform the row operation $R_{1} = R_{1} - R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 - 0 & 1 - 1 & 1 - 3 & 25 - 33 \\ 0 & 1 & 3 & 33 \end{array}]$

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & 0 & - 2 & - 8 \\ 0 & 1 & 3 & 33 \end{array}]$