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Question

# For ordinary arithmetic $2 \times 3=3 \times 2$ and in algebra ab = ba. For matrices, does AB always equal BA?

Solution

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$A= \quad \begin{bmatrix} 1 & 0 \\ 1 & 2 \end{bmatrix} \qquad \qquad B= \quad \begin{bmatrix} -1 & 1 \\ 0 & 3 \end{bmatrix} \qquad \qquad \text {Hint}$

$AB= \quad \begin{bmatrix} 1 & 0 \\ 1 & 2 \end{bmatrix} \begin{bmatrix} -1 & 1 \\ 0 & 3 \end{bmatrix} = \begin{bmatrix} -1 & 1 \\ -1 & 7 \end{bmatrix} \qquad \qquad \text {Multiply matrices}$

$BA= \quad \begin{bmatrix} -1 & 1 \\ 0 & 3 \end{bmatrix} \begin{bmatrix} 1 & 0 \\ 1 & 2 \end{bmatrix} = \begin{bmatrix} 0 & 2 \\ 3 & 6 \end{bmatrix} \qquad \qquad \text {Multiply matrices}$

$AB \neq BA$

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