Observation 5.4.2. Matrix Oddities.
Some of the main dissimilarities between matrix algebra and elementary algebra are that in matrix algebra:
- \(A B\) may be different from \(B A\text{.}\)
- There exist matrices \(A\) and \(B\) such that \(A B = \pmb{0}\text{,}\) and yet \(A\neq \pmb{0}\) and \(B\neq \pmb{0}\text{.}\)
- There exist matrices \(A\) where \(A \neq \pmb{0}\text{,}\) and yet \(A^2 = \pmb{0}\text{.}\)
- There exist matrices \(A\) where \(A^2=A\) with \(A\neq I\) and \(A\neq \pmb{0}\)
- There exist matrices \(A\) where \(A^2=I\text{,}\) where \(A\neq I\) and \(A\neq -I\)