Find the possible square roots of the two rowed unit matrix I.

Explanation

Let `A=[(a,b),(c,d)]` be a square root of the matrix `I`.
Then `A^(2)=I`, where `I=[(1,0),(0,1)]`
`implies [(a,b),(c,d)][(a,b),(c,d)]=[(1,0),(0,1)]`
`implies [(a^(2)+bc,ab+bd),(ac+cd,cb+d^(2))]=[(1,0),(0,1)]`
`implies a^(2)+bc=1` (1)
`ab+bd=0` (2)
`ac+cd=0` (3)
`cb+d^(2)=1` (4)
Case I : `a+d=0`
The above four equations hold simultaneously if `d=-a` and `a^(2)+bc=1`.
Hence, one possible square roots of `I` is `A=[(alpha,beta),(gamma,-alpha)]`, where `alpha, beta, gamma` are the three numbers related by the condition `alpha^(2)+beta gamma=1`.
Case II : `a+d+ ne 0`
The above four equation hold simultaneously
if `b=0, c=0, a=1, d=1` or if `b=0, c=0, a=-1, d=-1`
Hence, `[(1,0),(0,1)], [(-1,0),(0,-1)]` i.e., `pm I` are other possible equare roots of I.