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

1 Answer
Dec 7, 2016

Answer:

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) )#