How do you find the inverse of #A=##((1, 2, 1), (1, 2, -1), (-2, -2, -1)) #?

1 Answer
Alan P.
Dec 7, 2016

Augment the given matrix with identity matrix then reduce the augmented original matrix side to an identity matrix
to get the inverse (where the identity matrix was):
#color(white)("XXX")((-1,0,-1),(3/4,1/4,1/2),(1/2,-1/2,0))#

Explanation:

Initial Matrix Augmented By Identity Matrix:
#( (color(white)("XXX")1.000,color(white)("XXX")2.000,color(white)("XXX")1.000,color(white)("XXX")1.000,color(white)("XXX")0.000,color(white)("XXX")0.000), (color(white)("XXX")1.000,color(white)("XXX")2.000,color(white)("XXXX")-1.000,color(white)("XXX")0.000,color(white)("XXX")1.000,color(white)("XXX")0.000), (color(white)("XXXX")-2.000,color(white)("XXXX")-2.000,color(white)("XXXX")-1.000,color(white)("XXX")0.000,color(white)("XXX")0.000,color(white)("XXX")1.000) )#

Pivot Row: 1 Pivot Column: 1
pivot Row reduced:
#( (color(white)("XXX")1.000,color(white)("XXX")2.000,color(white)("XXX")1.000,color(white)("XXX")1.000,color(white)("XXX")0.000,color(white)("XXX")0.000), (color(white)("XXX")1.000,color(white)("XXX")2.000,color(white)("XXXX")-1.000,color(white)("XXX")0.000,color(white)("XXX")1.000,color(white)("XXX")0.000), (color(white)("XXXX")-2.000,color(white)("XXXX")-2.000,color(white)("XXXX")-1.000,color(white)("XXX")0.000,color(white)("XXX")0.000,color(white)("XXX")1.000) )#

Non-pivot rows reduced for pivot column
#( (color(white)("XXX")1.000,color(white)("XXX")2.000,color(white)("XXX")1.000,color(white)("XXX")1.000,color(white)("XXX")0.000,color(white)("XXX")0.000), (color(white)("XXX")0.000,color(white)("XXX")0.000,color(white)("XXXX")-2.000,color(white)("XXXX")-1.000,color(white)("XXX")1.000,color(white)("XXX")0.000), (color(white)("XXX")0.000,color(white)("XXX")2.000,color(white)("XXX")1.000,color(white)("XXX")2.000,color(white)("XXX")0.000,color(white)("XXX")1.000) )#

Pivot Row: 2 Pivot Column: 2
Swapped rows 2 and 3
#( (color(white)("XXX")1.000,color(white)("XXX")2.000,color(white)("XXX")1.000,color(white)("XXX")1.000,color(white)("XXX")0.000,color(white)("XXX")0.000), (color(white)("XXX")0.000,color(white)("XXX")2.000,color(white)("XXX")1.000,color(white)("XXX")2.000,color(white)("XXX")0.000,color(white)("XXX")1.000), (color(white)("XXX")0.000,color(white)("XXX")0.000,color(white)("XXXX")-2.000,color(white)("XXXX")-1.000,color(white)("XXX")1.000,color(white)("XXX")0.000) )#

pivot Row reduced:
#( (color(white)("XXX")1.000,color(white)("XXX")2.000,color(white)("XXX")1.000,color(white)("XXX")1.000,color(white)("XXX")0.000,color(white)("XXX")0.000), (color(white)("XXX")0.000,color(white)("XXX")1.000,color(white)("XXX")0.500,color(white)("XXX")1.000,color(white)("XXX")0.000,color(white)("XXX")0.500), (color(white)("XXX")0.000,color(white)("XXX")0.000,color(white)("XXXX")-2.000,color(white)("XXXX")-1.000,color(white)("XXX")1.000,color(white)("XXX")0.000) )#

Non-pivot rows reduced for pivot column
#( (color(white)("XXX")1.000,color(white)("XXX")0.000,color(white)("XXX")0.000,color(white)("XXXX")-1.000,color(white)("XXX")0.000,color(white)("XXXX")-1.000), (color(white)("XXX")0.000,color(white)("XXX")1.000,color(white)("XXX")0.500,color(white)("XXX")1.000,color(white)("XXX")0.000,color(white)("XXX")0.500), (color(white)("XXX")0.000,color(white)("XXX")0.000,color(white)("XXXX")-2.000,color(white)("XXXX")-1.000,color(white)("XXX")1.000,color(white)("XXX")0.000) )#

Pivot Row: 3 Pivot Column: 3
pivot Row reduced:
#( (color(white)("XXX")1.000,color(white)("XXX")0.000,color(white)("XXX")0.000,color(white)("XXXX")-1.000,color(white)("XXX")0.000,color(white)("XXXX")-1.000), (color(white)("XXX")0.000,color(white)("XXX")1.000,color(white)("XXX")0.500,color(white)("XXX")1.000,color(white)("XXX")0.000,color(white)("XXX")0.500), (color(white)("XXX")0.000,color(white)("XXX")0.000,color(white)("XXX")1.000,color(white)("XXX")0.500,color(white)("XXXX")-0.500,color(white)("XXX")0.000) )#

Non-pivot rows reduced for pivot column
#( (color(white)("XXX")1.000,color(white)("XXX")0.000,color(white)("XXX")0.000,color(white)("XXXX")-1.000,color(white)("XXX")0.000,color(white)("XXXX")-1.000), (color(white)("XXX")0.000,color(white)("XXX")1.000,color(white)("XXX")0.000,color(white)("XXX")0.750,color(white)("XXX")0.250,color(white)("XXX")0.500), (color(white)("XXX")0.000,color(white)("XXX")0.000,color(white)("XXX")1.000,color(white)("XXX")0.500,color(white)("XXXX")-0.500,color(white)("XXX")0.000) )#

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