 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2020/paper-1/3/c/solution

For

A_{1}

cos^{2} α_{1} - (i sin α_{1})^{2} = 1

. For

A_{2}

it is

cos^{2} α_{2} + sin^{2} α_{2} = 1

, and for

A_{3}

it is

e^{i α_{3}} e^{- i α_{3}} = 1

. Multiplicativity of the determinant gives

det A_{1} = det A_{2} = det A_{3} = 1, det (A_{1} A_{2} A_{3}) = 1.

Together with unitarity, this places the matrices and their product in SU(2).

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