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

1 Answer
Aug 27, 2016

Answer:

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

Explanation:

#A=((2, -1, 0, 0), (-1, 2, 0, 0), (0, 0, 2, -1), (0, 0, -1, 2))# is composed of two submatrices

#A = ((a,0_2),(0_2,a))#

with #a=((2,-1),(-1,2))#

and #0_2=((0,0),(0,0))#

so #A^(-1) = ((a^(-1),0_2),(0_2,a^(-1)))#

and

#a^(-1) = 1/3((2,1),(1,2)) #

so

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