Formulaic Maths Problems - get another problem: level 1, level 2, level 3 - link to this problem
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Let $A = \begin{pmatrix} 8 & -9 & 11 & 11 \\ 5 & 12 & -7 & -3 \\ -9 & 12 & 7 & -5 \\ 11 & 12 & -3 & -2 \end{pmatrix}$ and $B = \begin{pmatrix} -11 & -7 & 8 \\ 6 & -5 & 4 \\ -2 & 4 & -11 \\ 4 & 2 & -12 \end{pmatrix}$. Find $AB$.
Row 1, column 1: Hint: Row 1 of $A$ is $\begin{pmatrix} 8 & -9 & 11 & 11 \end{pmatrix}$. Column 1 of $B$ is $\begin{pmatrix} -11 \\ 6 \\ -2 \\ 4 \end{pmatrix}$. Multiply in pairs and add the results.
$\class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c0a0}}\hspace{40px}}~} \times \class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c0b0}}\hspace{40px}}~} + \class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c0a1}}\hspace{40px}}~} \times \class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c0b1}}\hspace{40px}}~} + \class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c0a2}}\hspace{40px}}~} \times \class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c0b2}}\hspace{40px}}~} + \class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c0a3}}\hspace{40px}}~} \times \class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c0b3}}\hspace{40px}}~}$ $[8] \times [-11] + [-9] \times [6] + [11] \times [-2] + [11] \times [4]$
$= \class{inputBox step2}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c0p0}}\hspace{54px}}~} + \class{inputBox step2}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c0p1}}\hspace{54px}}~} + \class{inputBox step2}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c0p2}}\hspace{54px}}~} + \class{inputBox step2}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c0p3}}\hspace{54px}}~} = \class{inputBox step2}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c0R}}\hspace{54px}}~}$ $= [-88] + [-54] + [-22] + [44] = [-120]$
Row 1, column 2: Hint: Row 1 of $A$ is $\begin{pmatrix} 8 & -9 & 11 & 11 \end{pmatrix}$. Column 2 of $B$ is $\begin{pmatrix} -7 \\ -5 \\ 4 \\ 2 \end{pmatrix}$. Multiply in pairs and add the results.
$\class{inputBox step3}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c1a0}}\hspace{40px}}~} \times \class{inputBox step3}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c1b0}}\hspace{40px}}~} + \class{inputBox step3}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c1a1}}\hspace{40px}}~} \times \class{inputBox step3}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c1b1}}\hspace{40px}}~} + \class{inputBox step3}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c1a2}}\hspace{40px}}~} \times \class{inputBox step3}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c1b2}}\hspace{40px}}~} + \class{inputBox step3}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c1a3}}\hspace{40px}}~} \times \class{inputBox step3}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c1b3}}\hspace{40px}}~}$ $[8] \times [-7] + [-9] \times [-5] + [11] \times [4] + [11] \times [2]$
$= \class{inputBox step4}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c1p0}}\hspace{54px}}~} + \class{inputBox step4}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c1p1}}\hspace{54px}}~} + \class{inputBox step4}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c1p2}}\hspace{54px}}~} + \class{inputBox step4}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c1p3}}\hspace{54px}}~} = \class{inputBox step4}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c1R}}\hspace{54px}}~}$ $= [-56] + [45] + [44] + [22] = [55]$
Row 1, column 3: Hint: Row 1 of $A$ is $\begin{pmatrix} 8 & -9 & 11 & 11 \end{pmatrix}$. Column 3 of $B$ is $\begin{pmatrix} 8 \\ 4 \\ -11 \\ -12 \end{pmatrix}$. Multiply in pairs and add the results.
$\class{inputBox step5}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c2a0}}\hspace{40px}}~} \times \class{inputBox step5}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c2b0}}\hspace{40px}}~} + \class{inputBox step5}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c2a1}}\hspace{40px}}~} \times \class{inputBox step5}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c2b1}}\hspace{40px}}~} + \class{inputBox step5}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c2a2}}\hspace{40px}}~} \times \class{inputBox step5}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c2b2}}\hspace{40px}}~} + \class{inputBox step5}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c2a3}}\hspace{40px}}~} \times \class{inputBox step5}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c2b3}}\hspace{40px}}~}$ $[8] \times [8] + [-9] \times [4] + [11] \times [-11] + [11] \times [-12]$
$= \class{inputBox step6}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c2p0}}\hspace{54px}}~} + \class{inputBox step6}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c2p1}}\hspace{54px}}~} + \class{inputBox step6}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c2p2}}\hspace{54px}}~} + \class{inputBox step6}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c2p3}}\hspace{54px}}~} = \class{inputBox step6}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c2R}}\hspace{54px}}~}$ $= [64] + [-36] + [-121] + [-132] = [-225]$
Row 2, column 1: Hint: Row 2 of $A$ is $\begin{pmatrix} 5 & 12 & -7 & -3 \end{pmatrix}$. Column 1 of $B$ is $\begin{pmatrix} -11 \\ 6 \\ -2 \\ 4 \end{pmatrix}$. Multiply in pairs and add the results.
$\class{inputBox step7}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c0a0}}\hspace{40px}}~} \times \class{inputBox step7}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c0b0}}\hspace{40px}}~} + \class{inputBox step7}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c0a1}}\hspace{40px}}~} \times \class{inputBox step7}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c0b1}}\hspace{40px}}~} + \class{inputBox step7}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c0a2}}\hspace{40px}}~} \times \class{inputBox step7}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c0b2}}\hspace{40px}}~} + \class{inputBox step7}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c0a3}}\hspace{40px}}~} \times \class{inputBox step7}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c0b3}}\hspace{40px}}~}$ $[5] \times [-11] + [12] \times [6] + [-7] \times [-2] + [-3] \times [4]$
$= \class{inputBox step8}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c0p0}}\hspace{54px}}~} + \class{inputBox step8}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c0p1}}\hspace{54px}}~} + \class{inputBox step8}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c0p2}}\hspace{54px}}~} + \class{inputBox step8}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c0p3}}\hspace{54px}}~} = \class{inputBox step8}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c0R}}\hspace{54px}}~}$ $= [-55] + [72] + [14] + [-12] = [19]$
Row 2, column 2: Hint: Row 2 of $A$ is $\begin{pmatrix} 5 & 12 & -7 & -3 \end{pmatrix}$. Column 2 of $B$ is $\begin{pmatrix} -7 \\ -5 \\ 4 \\ 2 \end{pmatrix}$. Multiply in pairs and add the results.
$\class{inputBox step9}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c1a0}}\hspace{40px}}~} \times \class{inputBox step9}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c1b0}}\hspace{40px}}~} + \class{inputBox step9}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c1a1}}\hspace{40px}}~} \times \class{inputBox step9}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c1b1}}\hspace{40px}}~} + \class{inputBox step9}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c1a2}}\hspace{40px}}~} \times \class{inputBox step9}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c1b2}}\hspace{40px}}~} + \class{inputBox step9}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c1a3}}\hspace{40px}}~} \times \class{inputBox step9}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c1b3}}\hspace{40px}}~}$ $[5] \times [-7] + [12] \times [-5] + [-7] \times [4] + [-3] \times [2]$
$= \class{inputBox step10}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c1p0}}\hspace{54px}}~} + \class{inputBox step10}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c1p1}}\hspace{54px}}~} + \class{inputBox step10}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c1p2}}\hspace{54px}}~} + \class{inputBox step10}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c1p3}}\hspace{54px}}~} = \class{inputBox step10}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c1R}}\hspace{54px}}~}$ $= [-35] + [-60] + [-28] + [-6] = [-129]$
Row 2, column 3: Hint: Row 2 of $A$ is $\begin{pmatrix} 5 & 12 & -7 & -3 \end{pmatrix}$. Column 3 of $B$ is $\begin{pmatrix} 8 \\ 4 \\ -11 \\ -12 \end{pmatrix}$. Multiply in pairs and add the results.
$\class{inputBox step11}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c2a0}}\hspace{40px}}~} \times \class{inputBox step11}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c2b0}}\hspace{40px}}~} + \class{inputBox step11}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c2a1}}\hspace{40px}}~} \times \class{inputBox step11}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c2b1}}\hspace{40px}}~} + \class{inputBox step11}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c2a2}}\hspace{40px}}~} \times \class{inputBox step11}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c2b2}}\hspace{40px}}~} + \class{inputBox step11}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c2a3}}\hspace{40px}}~} \times \class{inputBox step11}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c2b3}}\hspace{40px}}~}$ $[5] \times [8] + [12] \times [4] + [-7] \times [-11] + [-3] \times [-12]$
$= \class{inputBox step12}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c2p0}}\hspace{54px}}~} + \class{inputBox step12}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c2p1}}\hspace{54px}}~} + \class{inputBox step12}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c2p2}}\hspace{54px}}~} + \class{inputBox step12}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c2p3}}\hspace{54px}}~} = \class{inputBox step12}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c2R}}\hspace{54px}}~}$ $= [40] + [48] + [77] + [36] = [201]$
Row 3, column 1: Hint: Row 3 of $A$ is $\begin{pmatrix} -9 & 12 & 7 & -5 \end{pmatrix}$. Column 1 of $B$ is $\begin{pmatrix} -11 \\ 6 \\ -2 \\ 4 \end{pmatrix}$. Multiply in pairs and add the results.
$\class{inputBox step13}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c0a0}}\hspace{40px}}~} \times \class{inputBox step13}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c0b0}}\hspace{40px}}~} + \class{inputBox step13}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c0a1}}\hspace{40px}}~} \times \class{inputBox step13}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c0b1}}\hspace{40px}}~} + \class{inputBox step13}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c0a2}}\hspace{40px}}~} \times \class{inputBox step13}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c0b2}}\hspace{40px}}~} + \class{inputBox step13}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c0a3}}\hspace{40px}}~} \times \class{inputBox step13}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c0b3}}\hspace{40px}}~}$ $[-9] \times [-11] + [12] \times [6] + [7] \times [-2] + [-5] \times [4]$
$= \class{inputBox step14}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c0p0}}\hspace{54px}}~} + \class{inputBox step14}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c0p1}}\hspace{54px}}~} + \class{inputBox step14}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c0p2}}\hspace{54px}}~} + \class{inputBox step14}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c0p3}}\hspace{54px}}~} = \class{inputBox step14}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c0R}}\hspace{54px}}~}$ $= [99] + [72] + [-14] + [-20] = [137]$
Row 3, column 2: Hint: Row 3 of $A$ is $\begin{pmatrix} -9 & 12 & 7 & -5 \end{pmatrix}$. Column 2 of $B$ is $\begin{pmatrix} -7 \\ -5 \\ 4 \\ 2 \end{pmatrix}$. Multiply in pairs and add the results.
$\class{inputBox step15}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c1a0}}\hspace{40px}}~} \times \class{inputBox step15}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c1b0}}\hspace{40px}}~} + \class{inputBox step15}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c1a1}}\hspace{40px}}~} \times \class{inputBox step15}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c1b1}}\hspace{40px}}~} + \class{inputBox step15}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c1a2}}\hspace{40px}}~} \times \class{inputBox step15}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c1b2}}\hspace{40px}}~} + \class{inputBox step15}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c1a3}}\hspace{40px}}~} \times \class{inputBox step15}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c1b3}}\hspace{40px}}~}$ $[-9] \times [-7] + [12] \times [-5] + [7] \times [4] + [-5] \times [2]$
$= \class{inputBox step16}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c1p0}}\hspace{54px}}~} + \class{inputBox step16}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c1p1}}\hspace{54px}}~} + \class{inputBox step16}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c1p2}}\hspace{54px}}~} + \class{inputBox step16}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c1p3}}\hspace{54px}}~} = \class{inputBox step16}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c1R}}\hspace{54px}}~}$ $= [63] + [-60] + [28] + [-10] = [21]$
Row 3, column 3: Hint: Row 3 of $A$ is $\begin{pmatrix} -9 & 12 & 7 & -5 \end{pmatrix}$. Column 3 of $B$ is $\begin{pmatrix} 8 \\ 4 \\ -11 \\ -12 \end{pmatrix}$. Multiply in pairs and add the results.
$\class{inputBox step17}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c2a0}}\hspace{40px}}~} \times \class{inputBox step17}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c2b0}}\hspace{40px}}~} + \class{inputBox step17}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c2a1}}\hspace{40px}}~} \times \class{inputBox step17}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c2b1}}\hspace{40px}}~} + \class{inputBox step17}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c2a2}}\hspace{40px}}~} \times \class{inputBox step17}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c2b2}}\hspace{40px}}~} + \class{inputBox step17}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c2a3}}\hspace{40px}}~} \times \class{inputBox step17}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c2b3}}\hspace{40px}}~}$ $[-9] \times [8] + [12] \times [4] + [7] \times [-11] + [-5] \times [-12]$
$= \class{inputBox step18}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c2p0}}\hspace{54px}}~} + \class{inputBox step18}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c2p1}}\hspace{54px}}~} + \class{inputBox step18}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c2p2}}\hspace{54px}}~} + \class{inputBox step18}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c2p3}}\hspace{54px}}~} = \class{inputBox step18}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c2R}}\hspace{54px}}~}$ $= [-72] + [48] + [-77] + [60] = [-41]$
Row 4, column 1: Hint: Row 4 of $A$ is $\begin{pmatrix} 11 & 12 & -3 & -2 \end{pmatrix}$. Column 1 of $B$ is $\begin{pmatrix} -11 \\ 6 \\ -2 \\ 4 \end{pmatrix}$. Multiply in pairs and add the results.
$\class{inputBox step19}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c0a0}}\hspace{40px}}~} \times \class{inputBox step19}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c0b0}}\hspace{40px}}~} + \class{inputBox step19}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c0a1}}\hspace{40px}}~} \times \class{inputBox step19}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c0b1}}\hspace{40px}}~} + \class{inputBox step19}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c0a2}}\hspace{40px}}~} \times \class{inputBox step19}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c0b2}}\hspace{40px}}~} + \class{inputBox step19}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c0a3}}\hspace{40px}}~} \times \class{inputBox step19}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c0b3}}\hspace{40px}}~}$ $[11] \times [-11] + [12] \times [6] + [-3] \times [-2] + [-2] \times [4]$
$= \class{inputBox step20}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c0p0}}\hspace{54px}}~} + \class{inputBox step20}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c0p1}}\hspace{54px}}~} + \class{inputBox step20}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c0p2}}\hspace{54px}}~} + \class{inputBox step20}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c0p3}}\hspace{54px}}~} = \class{inputBox step20}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c0R}}\hspace{54px}}~}$ $= [-121] + [72] + [6] + [-8] = [-51]$
Row 4, column 2: Hint: Row 4 of $A$ is $\begin{pmatrix} 11 & 12 & -3 & -2 \end{pmatrix}$. Column 2 of $B$ is $\begin{pmatrix} -7 \\ -5 \\ 4 \\ 2 \end{pmatrix}$. Multiply in pairs and add the results.
$\class{inputBox step21}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c1a0}}\hspace{40px}}~} \times \class{inputBox step21}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c1b0}}\hspace{40px}}~} + \class{inputBox step21}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c1a1}}\hspace{40px}}~} \times \class{inputBox step21}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c1b1}}\hspace{40px}}~} + \class{inputBox step21}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c1a2}}\hspace{40px}}~} \times \class{inputBox step21}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c1b2}}\hspace{40px}}~} + \class{inputBox step21}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c1a3}}\hspace{40px}}~} \times \class{inputBox step21}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c1b3}}\hspace{40px}}~}$ $[11] \times [-7] + [12] \times [-5] + [-3] \times [4] + [-2] \times [2]$
$= \class{inputBox step22}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c1p0}}\hspace{54px}}~} + \class{inputBox step22}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c1p1}}\hspace{54px}}~} + \class{inputBox step22}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c1p2}}\hspace{54px}}~} + \class{inputBox step22}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c1p3}}\hspace{54px}}~} = \class{inputBox step22}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c1R}}\hspace{54px}}~}$ $= [-77] + [-60] + [-12] + [-4] = [-153]$
Row 4, column 3: Hint: Row 4 of $A$ is $\begin{pmatrix} 11 & 12 & -3 & -2 \end{pmatrix}$. Column 3 of $B$ is $\begin{pmatrix} 8 \\ 4 \\ -11 \\ -12 \end{pmatrix}$. Multiply in pairs and add the results.
$\class{inputBox step23}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c2a0}}\hspace{40px}}~} \times \class{inputBox step23}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c2b0}}\hspace{40px}}~} + \class{inputBox step23}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c2a1}}\hspace{40px}}~} \times \class{inputBox step23}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c2b1}}\hspace{40px}}~} + \class{inputBox step23}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c2a2}}\hspace{40px}}~} \times \class{inputBox step23}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c2b2}}\hspace{40px}}~} + \class{inputBox step23}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c2a3}}\hspace{40px}}~} \times \class{inputBox step23}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c2b3}}\hspace{40px}}~}$ $[11] \times [8] + [12] \times [4] + [-3] \times [-11] + [-2] \times [-12]$
$= \class{inputBox step24}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c2p0}}\hspace{54px}}~} + \class{inputBox step24}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c2p1}}\hspace{54px}}~} + \class{inputBox step24}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c2p2}}\hspace{54px}}~} + \class{inputBox step24}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c2p3}}\hspace{54px}}~} = \class{inputBox step24}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c2R}}\hspace{54px}}~}$ $= [88] + [48] + [33] + [24] = [193]$
Therefore,
$AB = \begin{pmatrix} \class{inputBox step25}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Fr0c0}}\hspace{54px}}~} & \class{inputBox step25}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Fr0c1}}\hspace{54px}}~} & \class{inputBox step25}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Fr0c2}}\hspace{54px}}~} \\ \class{inputBox step25}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Fr1c0}}\hspace{54px}}~} & \class{inputBox step25}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Fr1c1}}\hspace{54px}}~} & \class{inputBox step25}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Fr1c2}}\hspace{54px}}~} \\ \class{inputBox step25}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Fr2c0}}\hspace{54px}}~} & \class{inputBox step25}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Fr2c1}}\hspace{54px}}~} & \class{inputBox step25}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Fr2c2}}\hspace{54px}}~} \\ \class{inputBox step25}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Fr3c0}}\hspace{54px}}~} & \class{inputBox step25}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Fr3c1}}\hspace{54px}}~} & \class{inputBox step25}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Fr3c2}}\hspace{54px}}~} \end{pmatrix}$ $AB = \begin{pmatrix} [-120] & [55] & [-225] \\ [19] & [-129] & [201] \\ [137] & [21] & [-41] \\ [-51] & [-153] & [193] \end{pmatrix}$