The exponential of a square matrix $M$ is defined as

$$\exp M=I+\sum_{n=1}^{\infty} \frac{M^{n}}{n !}$$

where $I$ is the identity matrix. [You do not have to consider issues of convergence.]

(a) Calculate the elements of $R$ and $S$, where

$$R=\exp \left(\begin{array}{cc} 0 & -\theta \\ \theta & 0 \end{array}\right), \quad S=\exp \left(\begin{array}{ll} 0 & \theta \\ \theta & 0 \end{array}\right)$$

and $\theta$ is a real number.

(b) Show that $R R^{T}=I$ and that

$$S J S=J, \quad \text { where } \quad J=\left(\begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array}\right)$$

(c) Consider the matrices

$$A=\left(\begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & -1 / 2 \\ 0 & 1 / 2 & 0 \end{array}\right), \quad B=\left(\begin{array}{lll} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{array}\right)$$

Calculate:

(i) $\exp (x A)$,

(ii) $\exp (x B)$.

(d) Defining

$$C=\left(\begin{array}{ccc} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{array}\right)$$

find the elements of the following matrices, where $N$ is a natural number:

(i)

$$\sum_{n=1}^{N}\left(\exp (x A) C[\exp (x A)]^{T}\right)^{n}$$

(ii)

$$\sum_{n=1}^{N}(\exp (x B) C \exp (x B))^{n}$$

[Your answers to parts $(a),(c)$ and $(d)$ should be in closed form, i.e. not given as series.]