If (i) `A=[[cosalpha,sinalpha],[-sinalpha,cosalpha]]`, then verify that `AprimeA = I`.(ii) `A=[[sinalpha,cosalpha],[-cosalpha,sinalpha]]`, then verify that `AprimeA = I`.

Explanation

(i) Given `A=[[cosalpha,sinalpha],[-sinalpha,cosalpha]]`
`therefore`
`A'=[[cosalpha,-sinalpha],[sinalpha,cosalpha]]`
`A'A=[[cosalpha,-sinalpha],[sinalpha,cosalpha]] [[cosalpha,sinalpha],[-sinalpha,cosalpha]]`
`A'A=[[(cosalpha)(cosalpha)+(-sinalpha)(-sinalpha),(cosalpha)(sinalpha)+(-sinalpha)(cosalpha)],[(sinalpha)(cosalpha)+(cosalpha)(-sinalpha),(sinalpha)(sinalpha)+(cosalpha)(cosalpha)]] `
`A'A=[[cos^2alpha+sin^2alpha,sinalphacosalpha -sinalphacosalpha],[sinalphacosalpha-sinalphacosalpha,sin^2alpha+cos^2alpha ]] `
`A'A=[[1,0],[0,1]]=I`
Hence, we verified that `AprimeA = I`

(ii) `A=[[sinalpha,cosalpha],[-cosalpha,sinalpha]]`
`A'=[[sinalpha,-cosalpha],[cosalpha,sinalpha]]`
`A'A= [[sinalpha,-cosalpha],[cosalpha,sinalpha]][[sinalpha,cosalpha],[-cosalpha,sinalpha]]`

`A'A=[[(sinalpha)(sinalpha)+(-cosalpha)(-cosalpha),(sinalpha)(cosalpha)-(-cosalpha)(sinalpha)],[(sinalpha)(cosalpha)+(sinalpha)(-cosalpha),(cosalpha)(cosalpha)+(sinalpha)(sinalpha)]]`
`A'A=[[sin^2alpha+cos^2alpha,sinalphacosalpha -sinalphacosalpha],[sinalphacosalpha-sinalphacosalpha,cos^2alpha+sin^2alpha]] `

`A'A=[[1,0],[0,1]]=I`
Hence, we verified that `AprimeA = I`