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Question

# Compute the products $A$ $B$ and $B$ $A$, if possible, for the following:(a) $A$$=\left(\begin{array}{rr}0 & -2 \\ 3 & 1\end{array}\right), B$$=\left(\begin{array}{rr}-1 & 4 \\ 1 & 5\end{array}\right)$(b) $A$$=\left(\begin{array}{rrr}8 & 3 & -2 \\ 1 & 0 & 4\end{array}\right), B$$=\left(\begin{array}{rr}2 & -2 \\ 4 & 3 \\ 1 & -5\end{array}\right)$(c) $A$$=\left(\begin{array}{r}0 \\ -2 \\ 4\end{array}\right), B$$=\left(\begin{array}{lll}0, & -2, & 3\end{array}\right)$(d) $A$$=\left(\begin{array}{rr}-1 & 0 \\ 2 & 4\end{array}\right), \quad B$$=\left(\begin{array}{rr}3 & 1 \\ -1 & 1 \\ 0 & 2\end{array}\right)$

Solution

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(a) $\boldsymbol{A}\boldsymbol{B}$ is a $2\times2$ matrix obtained by multiplying each row of $\boldsymbol{A}$ by the corresponding column of $\boldsymbol{B}$.

\begin{aligned} \boldsymbol{A}\boldsymbol{B}&= \begin{pmatrix} 0&-2\\3&1 \end{pmatrix} \begin{pmatrix} -1&4\\1&5 \end{pmatrix}\\ &=\begin{pmatrix} 0\cdot(-1)-2\cdot1&0\cdot4-2\cdot5\\3\cdot(-1)+1\cdot1&3\cdot4+1\cdot5 \end{pmatrix}\\ &=\begin{pmatrix} -2 & -10\\-2&17 \end{pmatrix}\\ \end{aligned}

$\boldsymbol{B}\boldsymbol{A}$ is a $2\times2$ matrix obtained by multiplying each row of $\boldsymbol{B}$ by the corresponding column of $\boldsymbol{A}$.

\begin{aligned} \boldsymbol{B}\boldsymbol{A}&= \begin{pmatrix} -1&4\\1&5 \end{pmatrix} \begin{pmatrix} 0&-2\\3&1 \end{pmatrix}\\ &=\begin{pmatrix} -1\cdot 0+4\cdot3 & -1\cdot(-2)+4\cdot1\\1\cdot0+5\cdot3&1\cdot(-2)+5\cdot1 \end{pmatrix}\\ &=\begin{pmatrix} 12&6\\15&3 \end{pmatrix}\\ \end{aligned}

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