PlanetPhysics/Example of a Matrix Commutator

Calculate the [[../Commutator/|commutator]] $[A,B]$ of the [[../Matrix/|matrices]] $A=\left[\begin{array}{ccc}5& 3i& -i\\ 1& 0& 2\\ i& 2i& -1\end{array}\right]B=\left[\begin{array}{ccc}1& 0& 1\\ -2& 2i& 3\\ 3i& 1& -1\end{array}\right]$

Ans.

The commutator is calculated by definition as

$[A,B]=AB-BA$

Carrying out the first [[../Matrix/|matrix multiplication]] gives

$AB=\left[\begin{array}{ccc}8-6i& -6-i& 5+10i\\ 1+6i& 2& -1\\ -6i& -5& 1+7i\end{array}\right]$

and the second multiplication is

$BA=\left[\begin{array}{ccc}5+i& 5i& -1-i\\ -10+5i& 0& -3+6i\\ 1+14i& -9-2i& 6\end{array}\right].$

Finally, subtracting the two yields

$[A,B]=AB-BA\left[\begin{array}{ccc}3-7i& -6-6i& 6+11i\\ 11+i& 2& 2-6i\\ -1-20i& 4+2i& -5+7i\end{array}\right].$

Since the commutator is non-zero, we say that $A$ and $B$ do not [[../Commutator/|commute]].