$$\text { Prove or disprove that } \boldsymbol { A } \boldsymbol { B } = \boldsymbol { B } \boldsymbol { A } \text { whenever } \boldsymbol { A } \text { and } \boldsymbol { B } \text { are } 2 \times 2 \text { matrices. }$$

Question

Quizplus · Accepted Answer

The Answer is This is false. Counterexamples are easy to find. For instance, let

\boldsymbol { A } = \left[ \begin{array} { l l } 0 & 0 \ 2 & 0 \end{array} ight]

and

\boldsymbol { B } = \left[ \begin{array} { l l } 1 & 0 \ 1 & 0 \end{array} ight]

. Then

\boldsymbol { A } \boldsymbol { B } = \left[ \begin{array} { l l } 0 & 0 \ 2 & 0 \end{array} ight]

while

\boldsymbol { B } \boldsymbol { A } = \left[ \begin{array} { l l } 0 & 0 \ 0 & 0 \end{array} ight]