How do you solve $-x + 2y - 3z = 0$ , $2x + z = 0$ , and $3x - 4y + 4z = 2$ using matrices?

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Answer 1 · 2018-03-17T10:45:46

The solution is $S=((x=2/5),(y=-1),(z=-4/5))$

Perform the Gauss- Jordan elimination on the augmented matrix

$A=((-1,2,-3,|,0),(2,0,1,|,0),(3,-4,4,|,2))$

Make the pivot in the first column by

$R_1larr(-1)xxR_1$

$=((1,-2,3,|,0),(2,0,1,|,0),(3,-4,4,|,2))$

Eliminate the first column by

$R_2larrR_2-2R_1$ and $R_3larrR_3-3R_1$

$=((1,-2,3,|,0),(0,4,-5,|,0),(0,2,-5,|,2))$

Make the pivot in the second column by

$R_2larrR_2/4$

$=((1,-2,3,|,0),(0,1,-5/4,|,0),(0,2,-5,|,2))$

Eliminate the second column by

$R_3larrR_3-2R_2$ and $R_1larrR_1+2R_2$

$=((1,0,1/2,|,0),(0,1,-5/4,|,0),(0,0,-5/2,|,2))$

Make the pivot in the third column by

$R_3larrR_3/(-5/2)$

$=((1,0,1/2,|,0),(0,1,-5/4,|,0),(0,0,1,|,-4/5))$

Eliminate the third column by

$R_1larrR_1-1/2R_3$ and $R_2larrR_2+5/4R_3$

$=((1,0,0,|,2/5),(0,1,0,|,-1),(0,0,1,|,-4/5))$

The solution is

$S=((x=2/5),(y=-1),(z=-4/5))$

How do you solve -x + 2y - 3z = 0 -x + 2y - 3z = 0 , 2x + z = 0 2x + z = 0 , and 3x - 4y + 4z = 23x - 4y + 4z = 2 using matrices?