Question

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

Answer 1

`((1, 0, 0, 0),(0,0,0,1),(0,1,0,0),(0,0,1,0))`

Explanation:

`A = ((1, 0, 0, 0),(0,0,1,0),(0,0,0,1),(0,1,0,0))`

Notice what this matrix does as a linear transformation:

`((1, 0, 0, 0),(0,0,1,0),(0,0,0,1),(0,1,0,0))((a),(b),(c),(d)) = ((a),(c),(d),(b))`

It cyclically permutes three ordinates.

Therefore if applied `3` times it results in the identity:

`A^3 = I`

So the inverse of `A` is:

`A^2=((1, 0, 0, 0),(0,0,1,0),(0,0,0,1),(0,1,0,0))((1, 0, 0, 0),(0,0,1,0),(0,0,0,1),(0,1,0,0)) = ((1, 0, 0, 0),(0,0,0,1),(0,1,0,0),(0,0,1,0))`