Question

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

Answer 1

#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) )#