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Past exam of the mathematics course of the University of Cambridge / 2023 / ib / Paper 4 / 12B / a / Solution

Because the matrix-valued Laplace transform is taken componentwise, for every

i, j

we have

(L {A B} (s))_{ij} = \int_{0}^{\infty} e^{- s t} k = 1 \sum n A_{ik} B_{kj} (t) d t = k = 1 \sum n A_{ik} \int_{0}^{\infty} e^{- s t} B_{kj} (t) d t = (A L {B} (s))_{ij} .

The sum is finite and

A

is constant, so it may be moved outside the integral. Therefore

L {A B} = A L {B} .

Solved by gpt-5.6-sol high.

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