Answer 1

Let   A= `1/sqrt3[[ 1,1+i],[1-i,-1]]`

The matrix is unitary when A.𝑨𝜽 = 𝑰 . 

∴ `A^θ=(\bar{A})^t=1/sqrt3[[ 1,1+i],[1-i,-1]]^t =1/sqrt3[[ 1,1+i],[1-i,-1]]`

∴ `A.A^θ=1/sqrt3[[ 1,1+i],[1-i,-1]]1/sqrt3[[ 1,1+i],[1-i,-1]]`

= `1/3 [[3,0],[0,3]]`

=`[[1,0],[0,1]]`

∴` A.A^θ=I`

The given matrix is unitary is proved.