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))#