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Let $A = \begin{pmatrix} 10 & 2 \\ 3 & 1 \\ 1 & 8 \end{pmatrix}$ and $B = \begin{pmatrix} -10 & 8 & -3 & 11 \\ -11 & -4 & -12 & 10 \end{pmatrix}$. Find $AB$.
Row 1, column 1: Hint: Row 1 of $A$ is $\begin{pmatrix} 10 & 2 \end{pmatrix}$. Column 1 of $B$ is $\begin{pmatrix} -10 \\ -11 \end{pmatrix}$. Multiply in pairs and add the results.
$\class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c0a0}}\hspace{40px}}~} \times \class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c0b0}}\hspace{40px}}~} + \class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c0a1}}\hspace{40px}}~} \times \class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c0b1}}\hspace{40px}}~}$ $[10] \times [-10] + [2] \times [-11]$
$= \class{inputBox step2}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c0p0}}\hspace{54px}}~} + \class{inputBox step2}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c0p1}}\hspace{54px}}~} = \class{inputBox step2}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c0R}}\hspace{54px}}~}$ $= [-100] + [-22] = [-122]$
Row 1, column 2: Hint: Row 1 of $A$ is $\begin{pmatrix} 10 & 2 \end{pmatrix}$. Column 2 of $B$ is $\begin{pmatrix} 8 \\ -4 \end{pmatrix}$. Multiply in pairs and add the results.
$\class{inputBox step3}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c1a0}}\hspace{40px}}~} \times \class{inputBox step3}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c1b0}}\hspace{40px}}~} + \class{inputBox step3}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c1a1}}\hspace{40px}}~} \times \class{inputBox step3}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c1b1}}\hspace{40px}}~}$ $[10] \times [8] + [2] \times [-4]$
$= \class{inputBox step4}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c1p0}}\hspace{54px}}~} + \class{inputBox step4}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c1p1}}\hspace{54px}}~} = \class{inputBox step4}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c1R}}\hspace{54px}}~}$ $= [80] + [-8] = [72]$
Row 1, column 3: Hint: Row 1 of $A$ is $\begin{pmatrix} 10 & 2 \end{pmatrix}$. Column 3 of $B$ is $\begin{pmatrix} -3 \\ -12 \end{pmatrix}$. Multiply in pairs and add the results.
$\class{inputBox step5}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c2a0}}\hspace{40px}}~} \times \class{inputBox step5}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c2b0}}\hspace{40px}}~} + \class{inputBox step5}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c2a1}}\hspace{40px}}~} \times \class{inputBox step5}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c2b1}}\hspace{40px}}~}$ $[10] \times [-3] + [2] \times [-12]$
$= \class{inputBox step6}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c2p0}}\hspace{54px}}~} + \class{inputBox step6}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c2p1}}\hspace{54px}}~} = \class{inputBox step6}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c2R}}\hspace{54px}}~}$ $= [-30] + [-24] = [-54]$
Row 1, column 4: Hint: Row 1 of $A$ is $\begin{pmatrix} 10 & 2 \end{pmatrix}$. Column 4 of $B$ is $\begin{pmatrix} 11 \\ 10 \end{pmatrix}$. Multiply in pairs and add the results.
$\class{inputBox step7}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c3a0}}\hspace{40px}}~} \times \class{inputBox step7}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c3b0}}\hspace{40px}}~} + \class{inputBox step7}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c3a1}}\hspace{40px}}~} \times \class{inputBox step7}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c3b1}}\hspace{40px}}~}$ $[10] \times [11] + [2] \times [10]$
$= \class{inputBox step8}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c3p0}}\hspace{54px}}~} + \class{inputBox step8}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c3p1}}\hspace{54px}}~} = \class{inputBox step8}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c3R}}\hspace{54px}}~}$ $= [110] + [20] = [130]$
Row 2, column 1: Hint: Row 2 of $A$ is $\begin{pmatrix} 3 & 1 \end{pmatrix}$. Column 1 of $B$ is $\begin{pmatrix} -10 \\ -11 \end{pmatrix}$. Multiply in pairs and add the results.
$\class{inputBox step9}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c0a0}}\hspace{40px}}~} \times \class{inputBox step9}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c0b0}}\hspace{40px}}~} + \class{inputBox step9}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c0a1}}\hspace{40px}}~} \times \class{inputBox step9}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c0b1}}\hspace{40px}}~}$ $[3] \times [-10] + [1] \times [-11]$
$= \class{inputBox step10}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c0p0}}\hspace{54px}}~} + \class{inputBox step10}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c0p1}}\hspace{54px}}~} = \class{inputBox step10}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c0R}}\hspace{54px}}~}$ $= [-30] + [-11] = [-41]$
Row 2, column 2: Hint: Row 2 of $A$ is $\begin{pmatrix} 3 & 1 \end{pmatrix}$. Column 2 of $B$ is $\begin{pmatrix} 8 \\ -4 \end{pmatrix}$. Multiply in pairs and add the results.
$\class{inputBox step11}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c1a0}}\hspace{40px}}~} \times \class{inputBox step11}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c1b0}}\hspace{40px}}~} + \class{inputBox step11}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c1a1}}\hspace{40px}}~} \times \class{inputBox step11}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c1b1}}\hspace{40px}}~}$ $[3] \times [8] + [1] \times [-4]$
$= \class{inputBox step12}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c1p0}}\hspace{54px}}~} + \class{inputBox step12}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c1p1}}\hspace{54px}}~} = \class{inputBox step12}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c1R}}\hspace{54px}}~}$ $= [24] + [-4] = [20]$
Row 2, column 3: Hint: Row 2 of $A$ is $\begin{pmatrix} 3 & 1 \end{pmatrix}$. Column 3 of $B$ is $\begin{pmatrix} -3 \\ -12 \end{pmatrix}$. Multiply in pairs and add the results.
$\class{inputBox step13}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c2a0}}\hspace{40px}}~} \times \class{inputBox step13}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c2b0}}\hspace{40px}}~} + \class{inputBox step13}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c2a1}}\hspace{40px}}~} \times \class{inputBox step13}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c2b1}}\hspace{40px}}~}$ $[3] \times [-3] + [1] \times [-12]$
$= \class{inputBox step14}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c2p0}}\hspace{54px}}~} + \class{inputBox step14}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c2p1}}\hspace{54px}}~} = \class{inputBox step14}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c2R}}\hspace{54px}}~}$ $= [-9] + [-12] = [-21]$
Row 2, column 4: Hint: Row 2 of $A$ is $\begin{pmatrix} 3 & 1 \end{pmatrix}$. Column 4 of $B$ is $\begin{pmatrix} 11 \\ 10 \end{pmatrix}$. Multiply in pairs and add the results.
$\class{inputBox step15}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c3a0}}\hspace{40px}}~} \times \class{inputBox step15}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c3b0}}\hspace{40px}}~} + \class{inputBox step15}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c3a1}}\hspace{40px}}~} \times \class{inputBox step15}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c3b1}}\hspace{40px}}~}$ $[3] \times [11] + [1] \times [10]$
$= \class{inputBox step16}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c3p0}}\hspace{54px}}~} + \class{inputBox step16}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c3p1}}\hspace{54px}}~} = \class{inputBox step16}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c3R}}\hspace{54px}}~}$ $= [33] + [10] = [43]$
Row 3, column 1: Hint: Row 3 of $A$ is $\begin{pmatrix} 1 & 8 \end{pmatrix}$. Column 1 of $B$ is $\begin{pmatrix} -10 \\ -11 \end{pmatrix}$. Multiply in pairs and add the results.
$\class{inputBox step17}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c0a0}}\hspace{40px}}~} \times \class{inputBox step17}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c0b0}}\hspace{40px}}~} + \class{inputBox step17}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c0a1}}\hspace{40px}}~} \times \class{inputBox step17}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c0b1}}\hspace{40px}}~}$ $[1] \times [-10] + [8] \times [-11]$
$= \class{inputBox step18}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c0p0}}\hspace{54px}}~} + \class{inputBox step18}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c0p1}}\hspace{54px}}~} = \class{inputBox step18}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c0R}}\hspace{54px}}~}$ $= [-10] + [-88] = [-98]$
Row 3, column 2: Hint: Row 3 of $A$ is $\begin{pmatrix} 1 & 8 \end{pmatrix}$. Column 2 of $B$ is $\begin{pmatrix} 8 \\ -4 \end{pmatrix}$. Multiply in pairs and add the results.
$\class{inputBox step19}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c1a0}}\hspace{40px}}~} \times \class{inputBox step19}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c1b0}}\hspace{40px}}~} + \class{inputBox step19}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c1a1}}\hspace{40px}}~} \times \class{inputBox step19}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c1b1}}\hspace{40px}}~}$ $[1] \times [8] + [8] \times [-4]$
$= \class{inputBox step20}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c1p0}}\hspace{54px}}~} + \class{inputBox step20}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c1p1}}\hspace{54px}}~} = \class{inputBox step20}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c1R}}\hspace{54px}}~}$ $= [8] + [-32] = [-24]$
Row 3, column 3: Hint: Row 3 of $A$ is $\begin{pmatrix} 1 & 8 \end{pmatrix}$. Column 3 of $B$ is $\begin{pmatrix} -3 \\ -12 \end{pmatrix}$. Multiply in pairs and add the results.
$\class{inputBox step21}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c2a0}}\hspace{40px}}~} \times \class{inputBox step21}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c2b0}}\hspace{40px}}~} + \class{inputBox step21}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c2a1}}\hspace{40px}}~} \times \class{inputBox step21}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c2b1}}\hspace{40px}}~}$ $[1] \times [-3] + [8] \times [-12]$
$= \class{inputBox step22}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c2p0}}\hspace{54px}}~} + \class{inputBox step22}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c2p1}}\hspace{54px}}~} = \class{inputBox step22}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c2R}}\hspace{54px}}~}$ $= [-3] + [-96] = [-99]$
Row 3, column 4: Hint: Row 3 of $A$ is $\begin{pmatrix} 1 & 8 \end{pmatrix}$. Column 4 of $B$ is $\begin{pmatrix} 11 \\ 10 \end{pmatrix}$. Multiply in pairs and add the results.
$\class{inputBox step23}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c3a0}}\hspace{40px}}~} \times \class{inputBox step23}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c3b0}}\hspace{40px}}~} + \class{inputBox step23}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c3a1}}\hspace{40px}}~} \times \class{inputBox step23}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c3b1}}\hspace{40px}}~}$ $[1] \times [11] + [8] \times [10]$
$= \class{inputBox step24}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c3p0}}\hspace{54px}}~} + \class{inputBox step24}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c3p1}}\hspace{54px}}~} = \class{inputBox step24}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c3R}}\hspace{54px}}~}$ $= [11] + [80] = [91]$
Therefore,
$AB = \begin{pmatrix} \class{inputBox step25}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Fr0c0}}\hspace{54px}}~} & \class{inputBox step25}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Fr0c1}}\hspace{54px}}~} & \class{inputBox step25}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Fr0c2}}\hspace{54px}}~} & \class{inputBox step25}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Fr0c3}}\hspace{54px}}~} \\ \class{inputBox step25}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Fr1c0}}\hspace{54px}}~} & \class{inputBox step25}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Fr1c1}}\hspace{54px}}~} & \class{inputBox step25}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Fr1c2}}\hspace{54px}}~} & \class{inputBox step25}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Fr1c3}}\hspace{54px}}~} \\ \class{inputBox step25}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Fr2c0}}\hspace{54px}}~} & \class{inputBox step25}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Fr2c1}}\hspace{54px}}~} & \class{inputBox step25}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Fr2c2}}\hspace{54px}}~} & \class{inputBox step25}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Fr2c3}}\hspace{54px}}~} \end{pmatrix}$ $AB = \begin{pmatrix} [-122] & [72] & [-54] & [130] \\ [-41] & [20] & [-21] & [43] \\ [-98] & [-24] & [-99] & [91] \end{pmatrix}$