###### Observation5.4.2Matrix Oddities

Some of the main dissimilarities between matrix algebra and elementary algebra are that in matrix algebra:

\(A B\) may be different from \(B A\).

There exist matrices \(A\) and \(B\) such that \(A B = \pmb{0}\), and yet \(A\neq \pmb{0}\) and \(B\neq \pmb{0}\).

There exist matrices \(A\) where \(A \neq \pmb{0}\), and yet \(A^2 = \pmb{0}\).

There exist matrices \(A\) where \(A^2=A\) with \(A\neq I\) and \(A\neq \pmb{0}\)

There exist matrices \(A\) where \(A^2=I\), where \(A\neq I\) and \(A\neq -I\)