If `A=[a b0a]` is nth root of `I_2,` then choose the correct statements: If `n` is odd, `a=1,b=0` If `n` is odd, `a=-1,b=0` If `n` is even, `a=1,b=0` If `n` is even, `a=-1,b=0` a. i, ii, iii, iv b.ii, iii, iv c. i, ii, iii, iv d. i, iii, iv

Options

A. i, ii, iii
B. ii, iii, iv
C. i, ii, iii, iv
D. i, iii, iv

Answer

D

Explanation

If A is nth root of `I_(2)`, then `A^(n)=I_(2)`. Now,
`A^(2)=[(a,b),(0,a)][(a,b),(0,a)]=[(a^(2),2ab),(0,a^(2))]`
`A^(3)=A^(2) A=[(a^(2),2ab),(0,a^(2))][(a,b),(0,a)]=[(a^(3),3a^(2)b),(0,a^(3))]`
Thus, `A^(n)=[(n^(n),na^(n-1)b),(0,a^(n))]`
Now,
`A^(n)=I implies [(a^(n),na^(n-1)b),(0,a^(n))]=[(1,0),(0,1)]`
`implies a^(n)=1, b=0`