Solve the following systems of simultaneous equations using the inverse matrix of Gauss Jordan X1+2x2+3x3=3? 2x1+4x2+5x3=4? 3x1+5x2+6x3=8?

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Answer 1 · 2018-08-14T07:11:42

The solution is $((x_1),(x_2),(x_3))=((7),(-5),(2))$

The augmented matrix is

$A=((1,2,3,|,3),(2,4,5,|,4),(3,5,6,|,8))$

The main matrix is

$A_1=((1,2,3),(2,4,5),(3,5,6))$

The inverse is calculated as follows

Write side by side $A$ and $I_3$ on the right

$((1,2,3),(2,4,5),(3,5,6))((1,0,0),(0,1,0),(0,0,1))$

Perform the row operations

$R2larrR2-2xxR1$ and $R3larrR3-3xxR1$

$((1,2,3),(0,0,-1),(0,-1,-3))((1,0,0),(-2,1,0),(-3,0,1))$

$R3harrR2$

$((1,2,3),(0,-1,-3),(0,0,-1))((1,0,0),(-3,0,1),(-2,1,0))$

$R2larr(R2)/(-1)$ and $R3larr(R3)/(-1)$

$((1,2,3),(0,1,3),(0,0,1))((1,0,0),(3,0,-1),(2,-1,0))$

$R1larrR1-3xxR3$ and $R2larrR2-3xxR3$

$((1,2,0),(0,1,0),(0,0,1))((-5,3,0),(-3,3,-1),(2,-1,0))$

$R1larrR1-2xxR2$

$((1,0,0),(0,1,0),(0,0,1))((1,-3,2),(-3,3,-1),(2,-1,0))$

Therefore,

$A_1^-1=((1,-3,2),(-3,3,-1),(2,-1,0))$

Then,

$((x_1),(x_2),(x_3))=((1,-3,2),(-3,3,-1),(2,-1,0))*((3),(4),(8)$

$=((7),(-5),(2))$