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Bonus questions

MATH103 – Matrix methods - Lent 2014

Mark MacDonald

Here are a few bonus questions. Students who attempt these questions may hand their written (or typed) solutions to me at any time. Based on the clarity and correctness these solutions may earn them bonus marks (to be added to their coursework grade).

Furthermore, the final Exercise of every Section is to be considered a challenge question. I will consider giving extra credit to students who produce nice solutions to those.

Bonus 1:

Find distinct matrices $A,B,C\in M_{4}(\mathbb{Z})$ such that $A^{2}=B^{2}=C^{2}=-I_{2}$ , and $AB=-BA=C$ . Is the set $\{\pm I_{2},\pm A,\pm B,\pm C\}$ closed under multiplication? Justify your answer.
Bonus 2:
Define a map $[\cdot,\cdot]:M_{n}(\mathbb{R})\times M_{n}(\mathbb{R})\to M_{n}(\mathbb{R})$ as follows

$[A,B]:=AB-BA.$

This map is called the Lie bracket (pronounced “Lee”).
1. (a)
  
  Prove that $[A,B]=-[B,A]$
2. (b)
  
  Prove that $[A,[B,C]]+[B,[C,A]]+[C,[A,B]]=0$
3. (c)
  
  Find non-zero matrices $e,f,h\in M_{2}(\mathbb{R})$ such that
  
  $[e,f]=h,\quad[e,h]=2e,\quad[f,h]=2f.$
Bonus 3:

The centre of $M_{n}(\mathbb{R})$ is defined as the following set:

$Z(M_{n}(\mathbb{R})):=\{A\in M_{n}(\mathbb{R})\;|\;AB=BA\quad\hbox{{for all $B% \in M_{n}(\mathbb{R})$}}\quad\}.$

Prove that $Z(M_{n}(\mathbb{R}))=\{\lambda I_{n}\;|\;\lambda\in\mathbb{R}\}.$
Bonus 4:

Prove that if $A,B\in M_{n}(\mathbb{R})$ and $AB=I_{n}$ , then $BA=I_{n}$ .