Answer to Question #272693 in Linear Algebra for jackson

Question #272693

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


1
Expert's answer
2021-11-29T16:08:32-0500

1.

(300112109521252156330738108123121106)\begin{pmatrix} 300 & 112&109&|&521 \\ 252 & 156&330&|&738 \\ 108 & -123&121&|&106 \\ \end{pmatrix}


divide the 1 row by 300:


(128/75109/300521/300252156330738108123121106)\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:


(128/75109/300521/300061.92238.44300.360163.3281.7681.56)\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:


(128/75109/300521/300011987/5162503/5160163.3281.7681.56)\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:


(101663/1548115/1548011987/5162503/51600122235/172122235/172)\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:


(101663/1548115/1548011987/5162503/5160011)\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:


(100101010011)\begin{pmatrix} 1 & 0&0&|&1 \\ 0 & 1&0&|&1 \\ 0 &0&1&|&1 \\ \end{pmatrix}


x1=x2=x3=1x_1=x_2=x_3=1


2.

(111523584052)\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:


(1115013204118)\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:


(10270132001326)\begin{pmatrix} 1 & 0&-2&|&7 \\ 0 & 1&3&|&-2 \\ 0 & 0&13&|&-26 \\ \end{pmatrix}


divide the 3 row by 13:


(102701320012)\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:


(100301040012)\begin{pmatrix} 1 & 0&0&|&3 \\ 0 & 1&0&|&4 \\ 0 & 0&1&|&-2 \\ \end{pmatrix}


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


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