| course |
course_year |
question_number |
tags |
title |
year |
Linear Algebra |
IB |
2 |
|
2.I.1C |
2005 |
Let $\Omega$ be the set of all $2 \times 2$ matrices of the form $\alpha=a I+b J+c K+d L$, where $a, b, c, d$ are in $\mathbf{R}$, and
$$I=\left(\begin{array}{cc}
1 & 0 \\
0 & 1
\end{array}\right), J=\left(\begin{array}{cc}
i & 0 \\
0 & -i
\end{array}\right), K=\left(\begin{array}{cc}
0 & 1 \\
-1 & 0
\end{array}\right), L=\left(\begin{array}{cc}
0 & i \\
i & 0
\end{array}\right) \quad\left(i^{2}=-1\right) .$$
Prove that $\Omega$ is closed under multiplication and determine its dimension as a vector space over $\mathbf{R}$. Prove that
$$(a I+b J+c K+d L)(a I-b J-c K-d L)=\left(a^{2}+b^{2}+c^{2}+d^{2}\right) I$$
and deduce that each non-zero element of $\Omega$ is invertible.