How do you find the inverse of #A=##((0, 1, 2), (1, 3, 5), (-2, -3, -5))#?

1 Answer
Feb 16, 2016

Answer:

#A^(-1) = ((0,-1,-1),(-5,4,2),(3,-2,-1))#

Explanation:

Expand #A# with an identity matrix appended to its right side:

#((0,1,2,"|",1,0,0), (1,3,5,"|",0,1,0), (-2,-3,-5,"|",0,0,1))#

Our objective will be to perform standard row operations which will convert this matrix into a matrix with the identity matrix on the left side

Exchanging Row 1 and Row 2
#((1,3,5,"|",0,1,0), (0,1,2,"|",1,0,0), (-2,-3,-5,"|",0,0,1))#

Row 3 = (#2xx# Row 1) plus Row 3
#((1,3,5,"|",0,1,0), (0,1,2,"|",1,0,0), (0,3,5,"|",0,2,1))#

Row 1 = Row 1 minus (#3xx# Row 2)
#((1,0,-1,"|",-3,1,0), (0,1,2,"|",1,0,0), (0,3,5,"|",0,2,1))#

Row 3 = (#3xx# Row 2) minus Row 3
#((1,0,-1,"|",-3,1,0), (0,1,2,"|",1,0,0), (0,0,1,"|",3,-2,-1))#

Row 1 = Row 1 plus Row 3
#((1,0,0,"|",0,-1,-1), (0,1,2,"|",1,0,0), (0,0,1,"|",3,-2,-1))#

Row 2 = Row 2 minus (#2xx# Row 3)
#((1,0,0,"|",color(green)(0),color(green)(-1),color(green)(-1)), (0,1,0,"|",color(green)(-5),color(green)(4),color(green)(2)), (0,0,1,"|",color(green)(3),color(green)(-2),color(green)(-1)))#

The half of the matrix on the right side is the inverse of the original matrix.

You should check this result to ensure:
#A xx A^(-1) = I#
that is
#((0,1,2),(1,3,5),(-2,-3,-5))xx((0,-1,-1),(-5,4,2),(3,-2,-1)) = ((1,0,0),(0,1,0),(0,0,1))#