# Week 3

## Session 1

### Pre-class exercises

Exercise:

Prove that if $X$ if an $n$ -by-$n$ matrix with zeros everywhere except a 1 in the $ij$ th entry ($i\neq j$ ) then $\det(\exp(X))=\exp(\mathrm{Tr}(X))$ .

## Session 2

### Pre-class exercises

Exercise:

We constructed a homomorphism $R\colon SU(2)\to O(3)$ . Can you see why $R$ lands in the subset $SO(3)$ of matrices with determinant 1?

Exercise:

Can you prove that $\mathfrak{su}(2)$ is the set of 2-by-2 matrices $X$ such that $X^{T}=-X$ and $\mathrm{Tr}(X)=0$ ?