Question

#24, i am getting %5C%5C0%261%260%26%5Cfrac%7B8%7D%7B7%7D%20%5C%5C0%260%260%26%5Cfrac%7B2%7D%7B7%7D%20%5Cend%7Barray%7D%5Cright%5D" id="TexFormula1" title="\left[\begin{array}{cccc}1&0&0&\frac{18}{7} \\0&1&0&\frac{8}{7} \\0&0&0&\frac{2}{7} \end{array}\right]" alt="\left[\begin{array}{cccc}1&0&0&\frac{18}{7} \\0&1&0&\frac{8}{7} \\0&0&0&\frac{2}{7} \end{array}\right]" align="absmiddle" class="latex-formula"> , im wondering if someone can check it and show their work so i may compare.

Answer 1

It looks like you're talking about row-reducing an augmented matrix to solve the system of equations. Your answer is almost correct. The last row should read 0, 0, 1, 2/7.

The given system translates to

$\left[ \begin{array}{ccc|c} 2 & -3 & 1 & 2 \\ 1 & -1 & 2 & 2 \\ 1 & 2 & -3 & 4 \end{array} \right]$

Eliminate x from the last two rows by combining -2 (row 2) and row 1, and -2 (row 3) and row 1; that is,

(2x - 3y + z) - 2 (x - y + 2z) = 2 - 2 (2)

2x - 3y + z - 2x + 2y - 4z = 2 - 4

-y - 3z = -2

and

(2x - 3y + z) - 2 (x + 2y - 3z) = 2 - 2 (4)

2x - 3y + z - 2x - 4y + 6z = 2 - 8

-7y + 7z = -6

In augmented matrix form, this step yields

$\left[ \begin{array}{ccc|c} 2 & -3 & 1 & 2 \\ 0 & -1 & -3 & -2 \\ 0 & -7 & 7 & -6 \end{array} \right]$

I'll omit these details in the remaining steps.

Eliminate y from the last row by combining -7 (row 2) and row 3 :

$\left[ \begin{array}{ccc|c} 2 & -3 & 1 & 2 \\ 0 & -1 & -3 & -2 \\ 0 & 0 & 28 & 8 \end{array} \right]$

Multiply the last row by 1/28 :

$\left[ \begin{array}{ccc|c} 2 & -3 & 1 & 2 \\ 0 & -1 & -3 & -2 \\ 0 & 0 & 1 & 2/7 \end{array} \right]$

Eliminate z from the second row by combining 3 (row 3) and row 2 :

$\left[ \begin{array}{ccc|c} 2 & -3 & 1 & 2 \\ 0 & -1 & 0 & -8/7 \\ 0 & 0 & 1 & 2/7 \end{array} \right]$

Multiply the second row by -1 :

$\left[ \begin{array}{ccc|c} 2 & -3 & 1 & 2 \\ 0 & 1 & 0 & 8/7 \\ 0 & 0 & 1 & 2/7 \end{array} \right]$

Eliminate y and z from the first row by combining 3 (row 2), -1 (row 3), and row 1 :

$\left[ \begin{array}{ccc|c} 2 & 0 & 0 & 36/7 \\ 0 & 1 & 0 & 8/7 \\ 0 & 0 & 1 & 2/7 \end{array} \right]$

Multiply the first row by 1/2 :

$\left[ \begin{array}{ccc|c} 1 & 0 & 0 & 18/7 \\ 0 & 1 & 0 & 8/7 \\ 0 & 0 & 1 & 2/7 \end{array} \right]$