If the matrix `A=[{:(alpha, beta),(beta,alpha):}]` is such that `A^(2)=I`, then which one of the following is correct ?

Options

A. `alpha = 0, beta = 1 or alpha = 1, beta = 0`
B. `alpha = 0, beta ne 1 or alpha ne 1, beta = 1`
C. `alpha = 1, beta ne 0 or alpha ne 1, beta = 1
D. `alpha ne 0, beta ne 0`

Answer

A

Explanation

Let `A=[{:(alpha,beta),(beta,alpha):}]`
`therefore A^(2)=[{:(alpha,beta),(beta,alpha):}][{:(alpha,beta),(beta,alpha):}]`
`=[{:(alpha^(2)+beta^(2)," "2alpha beta),(" "2 alpha beta,alpha^(2)+beta^(2)):}]`
Now, `A^(2)=I`
`rArr [{:(alpha^(2)+beta^(2)," "2alpha beta),(" "2 alpha beta,alpha^(2)+beta^(2)):}]=[{:(1,0),(0,1):}]`
`rArr alpha^(2)+beta^(2)=1," "alpha beta = 0`
`rArr alpha = 0, beta = 1`
`or beta = 0, alpha = 1`