For the linear system with augmented matrix shown below, find the REF and then solve the system by back substitution. (Do not do any row exchanges. And do not use the ref command on your calculator because the calculator uses row exchanges and will?

Question

For the linear system with augmented matrix shown below, find the REF and then solve the system by back substitution. (Do not do any row exchanges. And do not use the ref command on your calculator because the calculator uses row exchanges and will generally give a different REF than is expected.)

4 −8 8 −16
8 −13 19 −23
12 −36 9 −75

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Answer 1 · 2017-09-20T11:01:22

The answer is #((x_1),(x_2),(x_3))=((14),(6),(-3))#

The augmented matrix is

#( (4,-8,8,|,-16), (8,-13,19,|,-23), (12,-36,9,|,-75) )#

Perform the following operations on the rows

#R2larrR2-2R1# and #R3larrR3-3R1#

#( (4,-8,8,|,-16), (0,3,3,|,9), (0,-12,-15,|,-27) )#

#R1larr(R1)/4# and #R2larr(R2)/3# and #(R3)/(-3)#

#( (1,-2,2,|,-4), (0,1,1,|,3), (0,4,5,|,9) )#

#R3larrR3-4R2#

#( (1,-2,2,|,-4), (0,1,1,|,3), (0,0,1,|,-3) )#

#R2larrR2-R3#

#( (1,-2,2,|,-4), (0,1,0,|,6), (0,0,1,|,-3) )#

#R1larrR1+2R2-R3))#

#( (1,0,0,|,14), (0,1,0,|,6), (0,0,1,|,-3) )#