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Prove that the matrix A=[[(1+i)/2,(-1+i)...

Prove that the matrix A=`[[(1+i)/2,(-1+i)/2],[(1+i)/2,(1-i)/2]]` is unitary.

Text Solution

Verified by Experts

`A^(theta)={:[((1-i)/(2),(1-i)/(2)),((-1-i)/(2),(1+i)/(2))]:}`
From `A^(theta)A={:[((1-i)/(2),(1-i)/(2)),((-1-i)/(2),(1+i)/(2))]{:[((1+i)/(2),(-1+i)/(2)),((1+i)/(2),(1-i)/(2))]:}`
`={:[((1-i^(2))/(4)+(1-i^(2))/(4),(-(1-i)^(2))/(4)+((1-i)^(2))/(4)),(-(1+i)^(2)/(4)+((1+i)^(2))/(4),(1-i^(2))/(4)+(1-i^(2))/(4))]:}`
`={:[((2)/(4)+(2)/(4),0),(0,(2)/(4)+(2)/(4))]:}={:[(1,0),(0,1)]:}=l`
Thus A is unitary
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