Paper 1, Section I, C

Vectors and Matrices | Part IA, 2017

Let AA and BB be real n×nn \times n matrices.

Show that (AB)T=BTAT(A B)^{T}=B^{T} A^{T}.

For any square matrix, the matrix exponential is defined by the series

eA=I+k=1Akk!e^{A}=I+\sum_{k=1}^{\infty} \frac{A^{k}}{k !}

Show that (eA)T=eAT\left(e^{A}\right)^{T}=e^{A^{T}}. [You are not required to consider issues of convergence.]

Calculate, in terms of AA and ATA^{T}, the matrices Q0,Q1Q_{0}, Q_{1} and Q2Q_{2} in the series for the matrix product

etAetAT=k=0Qktk, where tRe^{t A} e^{t A^{T}}=\sum_{k=0}^{\infty} Q_{k} t^{k}, \quad \text { where } t \in \mathbb{R}

Hence obtain a relation between AA and ATA^{T} which necessarily holds if etAe^{t A} is an orthogonal matrix.

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