Question

For the following linear system, put the augmented coefficient matrix into reduced row-echelon form.

2x1 + 3x2 − x3 = 14

x1 + 2x2 + x3 = 4

5x1 + 9x2 + 2x3 = 7

Answer 1

The augmented matrix of the system is
`\left[\begin{array}{cccc}2&3&-1&14\\1&2&1&4\\5&9&2&7\end{array}\right]`.

We apply operations rows:

1. We swap row 1 and 2 and obtain the matrix
`\left[\begin{array}{cccc}1&2&1&4\\2&3&-1&14\\5&9&2&7\end{array}\right]`.

2. Of the above matrix we subtract row 1 from row 2 twice (R2 - 2R1) and we subtract row 1 from row 3, 5 times. (R3-5R1). We obtain the matrix
`\left[\begin{array}{cccc}1&2&1&4\\0&-1&-3&6\\0&-1&-3&-13\end{array}\right]`

3. Of the above matrix we subtract row 2 from row1 twice (R3 - R2) and multiply the row 1 by -1 (-R2). Weobtain the matrix
`\left[\begin{array}{cccc}1&2&1&4\\0&1&3&-6\\0&0&0&-19\end{array}\right]`.

Since each pivote is an one then we conclude that the above matrix is the reduced row-echelon form of the matrix of the system.