How do you find the inverse of #A=##((2, -4), (1, 3))#?

1 Answer
Feb 16, 2016

Answer:

#A^(-1)=((3/10,2/5),(-1/10,1/5))#

Explanation:

Extend #A# with an identity matrix appended on its right:

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

Perform standard row operations with the objective of modifying the left side into an identity (sub)matrix.
(There are many ways to do this; the following is just one method).

Exchange Rows 1 and 2
#((1,3,"|",0,1),(2,-4,"|",1,0))#

Subtract #2xx# Row 1 from Row 2
#((1,3,"|",0,1),(0,-10,"|",1,-2))#

Divide Row 2 by #(-10)#
#((1,3,"|",0,1),(0,1,"|",-1/10,1/5))#

Subtract #3xx# Row 2 from Row 1
#((1,0,"|",3/10,2/5),(0,1,"|",-1/10,1/5))#

with the identity matrix on the left, the right side is the required inverse.

Of course, it's a good idea to verify this result:
#((2,-4),(1,3))xx((3/10,2/5),(-1/10,1/5))#

#=((6/10+4/10,color(white)("XXX"),4/5-4/5),(3/10-3/10,,2/5+3/5))#

#=((1,color(white)("XX"),0),(0,,1))#
...and we can go home happy!