Answer to Question #272693 in Linear Algebra for jackson

Question

1.      Use Gaussian elimination to solve the system of linear equations

300x₁ 112x₂ 109x₃ = 521

252x₁ 156x₂ 330x₃ =738

108x₁ -123x₂ 121x₃ =106

2.     Solve the following system linear equations by Gauss Jordan Method

x +y +z = 5

2x +3y +5z = 8

4x + 5z = 2

Answer 1

1.

$\begin{pmatrix} 300 & 112&109&|&521 \\ 252 & 156&330&|&738 \\ 108 & -123&121&|&106 \\ \end{pmatrix}$

divide the 1 row by 300:

$\begin{pmatrix} 1 & 28/75&109/300&|&521/300 \\ 252 & 156&330&|&738 \\ 108 & -123&121&|&106 \\ \end{pmatrix}$

multiply 1 row by 252 and subtract it from 2 row; multiply 1 row by 108 and subtract it from 3 row:

$\begin{pmatrix} 1 & 28/75&109/300&|&521/300 \\ 0 & 61.92&238.44&|&300.36 \\ 0 & -163.32&81.76&|&-81.56 \\ \end{pmatrix}$

divide the 2 row by 61.92:

$\begin{pmatrix} 1 & 28/75&109/300&|&521/300 \\ 0 & 1&1987/516&|&2503/516 \\ 0 & -163.32&81.76&|&-81.56 \\ \end{pmatrix}$

multiply 2 row by 28/75 and subtract it from 1 row; (multiply 2 row by 163.32 and add it to 3 row:

$\begin{pmatrix} 1 & 0&-1663/1548&|&-115/1548 \\ 0 & 1&1987/516&|&2503/516 \\ 0 &0&122235/172&|&122235/172 \\ \end{pmatrix}$

divide the 3 row by 122235/172:

$\begin{pmatrix} 1 & 0&-1663/1548&|&-115/1548 \\ 0 & 1&1987/516&|&2503/516 \\ 0 &0&1&|&1 \\ \end{pmatrix}$

(multiply 3 row by 1663/1548 and add it to 1 row; multiply 3 row by 1987/516 and subtract it from 2 row:

$\begin{pmatrix} 1 & 0&0&|&1 \\ 0 & 1&0&|&1 \\ 0 &0&1&|&1 \\ \end{pmatrix}$

$x_1=x_2=x_3=1$

2.

$\begin{pmatrix} 1 & 1&1&|&5 \\ 2 & 3&5&|&8 \\ 4 & 0&5&|&2 \\ \end{pmatrix}$

multiply 1 row by 2 and subtract it from 2 row; multiply 1 row by 4 and subtract it from 3 row:

$\begin{pmatrix} 1 & 1&1&|&5 \\ 0 & 1&3&|&-2 \\ 0 & -4&1&|&-18 \\ \end{pmatrix}$

multiply 2 row by 1 and subtract it from 1 row; multiply 2 row by 4 and add it to 3 row:

$\begin{pmatrix} 1 & 0&-2&|&7 \\ 0 & 1&3&|&-2 \\ 0 & 0&13&|&-26 \\ \end{pmatrix}$

divide the 3 row by 13:

$\begin{pmatrix} 1 & 0&-2&|&7 \\ 0 & 1&3&|&-2 \\ 0 & 0&1&|&-2 \\ \end{pmatrix}$

multiply 3 row by 2 and add it to 1 row; multiply 3 row by 3 and subtract it from 2 row:

$\begin{pmatrix} 1 & 0&0&|&3 \\ 0 & 1&0&|&4 \\ 0 & 0&1&|&-2 \\ \end{pmatrix}$

$x=3,y=4,z=-2$