 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2021/paper-2/17/c/i/solution

Let

Y = (p r q s)

. The commutativity condition gives

X Y = (2 r - p 2 s - q), Y X = (- q - s 2 p 2 r) .

Equality is equivalent to

s = p

and

q = - 2 r

. Therefore

Y = (α - β 2 β α) = α I + βX, α, β \in R .

Each displayed matrix commutes with

X

, since it is a polynomial in

X

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