Formulaic Maths Problems - get another problem: level 1, level 2, level 3 - link to this problem
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Let $A = \begin{pmatrix} -5 & -2 & 8 & 1 \\ 8 & -3 & 12 & 3 \\ -1 & -9 & 1 & 12 \\ -5 & -1 & -12 & 4 \end{pmatrix}$ and $B = \begin{pmatrix} 5 & -9 & 0 & 10 \\ -8 & 3 & 8 & -11 \\ 4 & -7 & -10 & -12 \\ 0 & -8 & -11 & 5 \end{pmatrix}$. Find $AB$.
Row 1, column 1: Hint: Row 1 of $A$ is $\begin{pmatrix} -5 & -2 & 8 & 1 \end{pmatrix}$. Column 1 of $B$ is $\begin{pmatrix} 5 \\ -8 \\ 4 \\ 0 \end{pmatrix}$. Multiply in pairs and add the results.
$\class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c0a0}}\hspace{40px}}~} \times \class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c0b0}}\hspace{40px}}~} + \class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c0a1}}\hspace{40px}}~} \times \class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c0b1}}\hspace{40px}}~} + \class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c0a2}}\hspace{40px}}~} \times \class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c0b2}}\hspace{40px}}~} + \class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c0a3}}\hspace{40px}}~} \times \class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c0b3}}\hspace{40px}}~}$ $[-5] \times [5] + [-2] \times [-8] + [8] \times [4] + [1] \times [0]$
$= \class{inputBox step2}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c0p0}}\hspace{54px}}~} + \class{inputBox step2}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c0p1}}\hspace{54px}}~} + \class{inputBox step2}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c0p2}}\hspace{54px}}~} + \class{inputBox step2}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c0p3}}\hspace{54px}}~} = \class{inputBox step2}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c0R}}\hspace{54px}}~}$ $= [-25] + [16] + [32] + [0] = [23]$
Row 1, column 2: Hint: Row 1 of $A$ is $\begin{pmatrix} -5 & -2 & 8 & 1 \end{pmatrix}$. Column 2 of $B$ is $\begin{pmatrix} -9 \\ 3 \\ -7 \\ -8 \end{pmatrix}$. Multiply in pairs and add the results.
$\class{inputBox step3}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c1a0}}\hspace{40px}}~} \times \class{inputBox step3}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c1b0}}\hspace{40px}}~} + \class{inputBox step3}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c1a1}}\hspace{40px}}~} \times \class{inputBox step3}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c1b1}}\hspace{40px}}~} + \class{inputBox step3}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c1a2}}\hspace{40px}}~} \times \class{inputBox step3}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c1b2}}\hspace{40px}}~} + \class{inputBox step3}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c1a3}}\hspace{40px}}~} \times \class{inputBox step3}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c1b3}}\hspace{40px}}~}$ $[-5] \times [-9] + [-2] \times [3] + [8] \times [-7] + [1] \times [-8]$
$= \class{inputBox step4}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c1p0}}\hspace{54px}}~} + \class{inputBox step4}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c1p1}}\hspace{54px}}~} + \class{inputBox step4}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c1p2}}\hspace{54px}}~} + \class{inputBox step4}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c1p3}}\hspace{54px}}~} = \class{inputBox step4}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c1R}}\hspace{54px}}~}$ $= [45] + [-6] + [-56] + [-8] = [-25]$
Row 1, column 3: Hint: Row 1 of $A$ is $\begin{pmatrix} -5 & -2 & 8 & 1 \end{pmatrix}$. Column 3 of $B$ is $\begin{pmatrix} 0 \\ 8 \\ -10 \\ -11 \end{pmatrix}$. Multiply in pairs and add the results.
$\class{inputBox step5}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c2a0}}\hspace{40px}}~} \times \class{inputBox step5}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c2b0}}\hspace{40px}}~} + \class{inputBox step5}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c2a1}}\hspace{40px}}~} \times \class{inputBox step5}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c2b1}}\hspace{40px}}~} + \class{inputBox step5}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c2a2}}\hspace{40px}}~} \times \class{inputBox step5}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c2b2}}\hspace{40px}}~} + \class{inputBox step5}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c2a3}}\hspace{40px}}~} \times \class{inputBox step5}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c2b3}}\hspace{40px}}~}$ $[-5] \times [0] + [-2] \times [8] + [8] \times [-10] + [1] \times [-11]$
$= \class{inputBox step6}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c2p0}}\hspace{54px}}~} + \class{inputBox step6}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c2p1}}\hspace{54px}}~} + \class{inputBox step6}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c2p2}}\hspace{54px}}~} + \class{inputBox step6}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c2p3}}\hspace{54px}}~} = \class{inputBox step6}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c2R}}\hspace{54px}}~}$ $= [0] + [-16] + [-80] + [-11] = [-107]$
Row 1, column 4: Hint: Row 1 of $A$ is $\begin{pmatrix} -5 & -2 & 8 & 1 \end{pmatrix}$. Column 4 of $B$ is $\begin{pmatrix} 10 \\ -11 \\ -12 \\ 5 \end{pmatrix}$. Multiply in pairs and add the results.
$\class{inputBox step7}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c3a0}}\hspace{40px}}~} \times \class{inputBox step7}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c3b0}}\hspace{40px}}~} + \class{inputBox step7}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c3a1}}\hspace{40px}}~} \times \class{inputBox step7}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c3b1}}\hspace{40px}}~} + \class{inputBox step7}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c3a2}}\hspace{40px}}~} \times \class{inputBox step7}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c3b2}}\hspace{40px}}~} + \class{inputBox step7}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c3a3}}\hspace{40px}}~} \times \class{inputBox step7}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c3b3}}\hspace{40px}}~}$ $[-5] \times [10] + [-2] \times [-11] + [8] \times [-12] + [1] \times [5]$
$= \class{inputBox step8}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c3p0}}\hspace{54px}}~} + \class{inputBox step8}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c3p1}}\hspace{54px}}~} + \class{inputBox step8}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c3p2}}\hspace{54px}}~} + \class{inputBox step8}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c3p3}}\hspace{54px}}~} = \class{inputBox step8}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c3R}}\hspace{54px}}~}$ $= [-50] + [22] + [-96] + [5] = [-119]$
Row 2, column 1: Hint: Row 2 of $A$ is $\begin{pmatrix} 8 & -3 & 12 & 3 \end{pmatrix}$. Column 1 of $B$ is $\begin{pmatrix} 5 \\ -8 \\ 4 \\ 0 \end{pmatrix}$. Multiply in pairs and add the results.
$\class{inputBox step9}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c0a0}}\hspace{40px}}~} \times \class{inputBox step9}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c0b0}}\hspace{40px}}~} + \class{inputBox step9}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c0a1}}\hspace{40px}}~} \times \class{inputBox step9}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c0b1}}\hspace{40px}}~} + \class{inputBox step9}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c0a2}}\hspace{40px}}~} \times \class{inputBox step9}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c0b2}}\hspace{40px}}~} + \class{inputBox step9}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c0a3}}\hspace{40px}}~} \times \class{inputBox step9}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c0b3}}\hspace{40px}}~}$ $[8] \times [5] + [-3] \times [-8] + [12] \times [4] + [3] \times [0]$
$= \class{inputBox step10}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c0p0}}\hspace{54px}}~} + \class{inputBox step10}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c0p1}}\hspace{54px}}~} + \class{inputBox step10}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c0p2}}\hspace{54px}}~} + \class{inputBox step10}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c0p3}}\hspace{54px}}~} = \class{inputBox step10}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c0R}}\hspace{54px}}~}$ $= [40] + [24] + [48] + [0] = [112]$
Row 2, column 2: Hint: Row 2 of $A$ is $\begin{pmatrix} 8 & -3 & 12 & 3 \end{pmatrix}$. Column 2 of $B$ is $\begin{pmatrix} -9 \\ 3 \\ -7 \\ -8 \end{pmatrix}$. Multiply in pairs and add the results.
$\class{inputBox step11}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c1a0}}\hspace{40px}}~} \times \class{inputBox step11}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c1b0}}\hspace{40px}}~} + \class{inputBox step11}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c1a1}}\hspace{40px}}~} \times \class{inputBox step11}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c1b1}}\hspace{40px}}~} + \class{inputBox step11}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c1a2}}\hspace{40px}}~} \times \class{inputBox step11}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c1b2}}\hspace{40px}}~} + \class{inputBox step11}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c1a3}}\hspace{40px}}~} \times \class{inputBox step11}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c1b3}}\hspace{40px}}~}$ $[8] \times [-9] + [-3] \times [3] + [12] \times [-7] + [3] \times [-8]$
$= \class{inputBox step12}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c1p0}}\hspace{54px}}~} + \class{inputBox step12}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c1p1}}\hspace{54px}}~} + \class{inputBox step12}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c1p2}}\hspace{54px}}~} + \class{inputBox step12}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c1p3}}\hspace{54px}}~} = \class{inputBox step12}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c1R}}\hspace{54px}}~}$ $= [-72] + [-9] + [-84] + [-24] = [-189]$
Row 2, column 3: Hint: Row 2 of $A$ is $\begin{pmatrix} 8 & -3 & 12 & 3 \end{pmatrix}$. Column 3 of $B$ is $\begin{pmatrix} 0 \\ 8 \\ -10 \\ -11 \end{pmatrix}$. Multiply in pairs and add the results.
$\class{inputBox step13}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c2a0}}\hspace{40px}}~} \times \class{inputBox step13}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c2b0}}\hspace{40px}}~} + \class{inputBox step13}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c2a1}}\hspace{40px}}~} \times \class{inputBox step13}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c2b1}}\hspace{40px}}~} + \class{inputBox step13}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c2a2}}\hspace{40px}}~} \times \class{inputBox step13}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c2b2}}\hspace{40px}}~} + \class{inputBox step13}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c2a3}}\hspace{40px}}~} \times \class{inputBox step13}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c2b3}}\hspace{40px}}~}$ $[8] \times [0] + [-3] \times [8] + [12] \times [-10] + [3] \times [-11]$
$= \class{inputBox step14}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c2p0}}\hspace{54px}}~} + \class{inputBox step14}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c2p1}}\hspace{54px}}~} + \class{inputBox step14}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c2p2}}\hspace{54px}}~} + \class{inputBox step14}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c2p3}}\hspace{54px}}~} = \class{inputBox step14}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c2R}}\hspace{54px}}~}$ $= [0] + [-24] + [-120] + [-33] = [-177]$
Row 2, column 4: Hint: Row 2 of $A$ is $\begin{pmatrix} 8 & -3 & 12 & 3 \end{pmatrix}$. Column 4 of $B$ is $\begin{pmatrix} 10 \\ -11 \\ -12 \\ 5 \end{pmatrix}$. Multiply in pairs and add the results.
$\class{inputBox step15}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c3a0}}\hspace{40px}}~} \times \class{inputBox step15}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c3b0}}\hspace{40px}}~} + \class{inputBox step15}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c3a1}}\hspace{40px}}~} \times \class{inputBox step15}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c3b1}}\hspace{40px}}~} + \class{inputBox step15}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c3a2}}\hspace{40px}}~} \times \class{inputBox step15}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c3b2}}\hspace{40px}}~} + \class{inputBox step15}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c3a3}}\hspace{40px}}~} \times \class{inputBox step15}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c3b3}}\hspace{40px}}~}$ $[8] \times [10] + [-3] \times [-11] + [12] \times [-12] + [3] \times [5]$
$= \class{inputBox step16}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c3p0}}\hspace{54px}}~} + \class{inputBox step16}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c3p1}}\hspace{54px}}~} + \class{inputBox step16}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c3p2}}\hspace{54px}}~} + \class{inputBox step16}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c3p3}}\hspace{54px}}~} = \class{inputBox step16}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c3R}}\hspace{54px}}~}$ $= [80] + [33] + [-144] + [15] = [-16]$
Row 3, column 1: Hint: Row 3 of $A$ is $\begin{pmatrix} -1 & -9 & 1 & 12 \end{pmatrix}$. Column 1 of $B$ is $\begin{pmatrix} 5 \\ -8 \\ 4 \\ 0 \end{pmatrix}$. Multiply in pairs and add the results.
$\class{inputBox step17}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c0a0}}\hspace{40px}}~} \times \class{inputBox step17}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c0b0}}\hspace{40px}}~} + \class{inputBox step17}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c0a1}}\hspace{40px}}~} \times \class{inputBox step17}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c0b1}}\hspace{40px}}~} + \class{inputBox step17}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c0a2}}\hspace{40px}}~} \times \class{inputBox step17}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c0b2}}\hspace{40px}}~} + \class{inputBox step17}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c0a3}}\hspace{40px}}~} \times \class{inputBox step17}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c0b3}}\hspace{40px}}~}$ $[-1] \times [5] + [-9] \times [-8] + [1] \times [4] + [12] \times [0]$
$= \class{inputBox step18}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c0p0}}\hspace{54px}}~} + \class{inputBox step18}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c0p1}}\hspace{54px}}~} + \class{inputBox step18}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c0p2}}\hspace{54px}}~} + \class{inputBox step18}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c0p3}}\hspace{54px}}~} = \class{inputBox step18}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c0R}}\hspace{54px}}~}$ $= [-5] + [72] + [4] + [0] = [71]$
Row 3, column 2: Hint: Row 3 of $A$ is $\begin{pmatrix} -1 & -9 & 1 & 12 \end{pmatrix}$. Column 2 of $B$ is $\begin{pmatrix} -9 \\ 3 \\ -7 \\ -8 \end{pmatrix}$. Multiply in pairs and add the results.
$\class{inputBox step19}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c1a0}}\hspace{40px}}~} \times \class{inputBox step19}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c1b0}}\hspace{40px}}~} + \class{inputBox step19}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c1a1}}\hspace{40px}}~} \times \class{inputBox step19}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c1b1}}\hspace{40px}}~} + \class{inputBox step19}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c1a2}}\hspace{40px}}~} \times \class{inputBox step19}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c1b2}}\hspace{40px}}~} + \class{inputBox step19}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c1a3}}\hspace{40px}}~} \times \class{inputBox step19}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c1b3}}\hspace{40px}}~}$ $[-1] \times [-9] + [-9] \times [3] + [1] \times [-7] + [12] \times [-8]$
$= \class{inputBox step20}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c1p0}}\hspace{54px}}~} + \class{inputBox step20}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c1p1}}\hspace{54px}}~} + \class{inputBox step20}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c1p2}}\hspace{54px}}~} + \class{inputBox step20}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c1p3}}\hspace{54px}}~} = \class{inputBox step20}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c1R}}\hspace{54px}}~}$ $= [9] + [-27] + [-7] + [-96] = [-121]$
Row 3, column 3: Hint: Row 3 of $A$ is $\begin{pmatrix} -1 & -9 & 1 & 12 \end{pmatrix}$. Column 3 of $B$ is $\begin{pmatrix} 0 \\ 8 \\ -10 \\ -11 \end{pmatrix}$. Multiply in pairs and add the results.
$\class{inputBox step21}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c2a0}}\hspace{40px}}~} \times \class{inputBox step21}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c2b0}}\hspace{40px}}~} + \class{inputBox step21}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c2a1}}\hspace{40px}}~} \times \class{inputBox step21}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c2b1}}\hspace{40px}}~} + \class{inputBox step21}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c2a2}}\hspace{40px}}~} \times \class{inputBox step21}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c2b2}}\hspace{40px}}~} + \class{inputBox step21}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c2a3}}\hspace{40px}}~} \times \class{inputBox step21}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c2b3}}\hspace{40px}}~}$ $[-1] \times [0] + [-9] \times [8] + [1] \times [-10] + [12] \times [-11]$
$= \class{inputBox step22}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c2p0}}\hspace{54px}}~} + \class{inputBox step22}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c2p1}}\hspace{54px}}~} + \class{inputBox step22}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c2p2}}\hspace{54px}}~} + \class{inputBox step22}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c2p3}}\hspace{54px}}~} = \class{inputBox step22}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c2R}}\hspace{54px}}~}$ $= [0] + [-72] + [-10] + [-132] = [-214]$
Row 3, column 4: Hint: Row 3 of $A$ is $\begin{pmatrix} -1 & -9 & 1 & 12 \end{pmatrix}$. Column 4 of $B$ is $\begin{pmatrix} 10 \\ -11 \\ -12 \\ 5 \end{pmatrix}$. Multiply in pairs and add the results.
$\class{inputBox step23}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c3a0}}\hspace{40px}}~} \times \class{inputBox step23}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c3b0}}\hspace{40px}}~} + \class{inputBox step23}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c3a1}}\hspace{40px}}~} \times \class{inputBox step23}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c3b1}}\hspace{40px}}~} + \class{inputBox step23}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c3a2}}\hspace{40px}}~} \times \class{inputBox step23}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c3b2}}\hspace{40px}}~} + \class{inputBox step23}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c3a3}}\hspace{40px}}~} \times \class{inputBox step23}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c3b3}}\hspace{40px}}~}$ $[-1] \times [10] + [-9] \times [-11] + [1] \times [-12] + [12] \times [5]$
$= \class{inputBox step24}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c3p0}}\hspace{54px}}~} + \class{inputBox step24}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c3p1}}\hspace{54px}}~} + \class{inputBox step24}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c3p2}}\hspace{54px}}~} + \class{inputBox step24}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c3p3}}\hspace{54px}}~} = \class{inputBox step24}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c3R}}\hspace{54px}}~}$ $= [-10] + [99] + [-12] + [60] = [137]$
Row 4, column 1: Hint: Row 4 of $A$ is $\begin{pmatrix} -5 & -1 & -12 & 4 \end{pmatrix}$. Column 1 of $B$ is $\begin{pmatrix} 5 \\ -8 \\ 4 \\ 0 \end{pmatrix}$. Multiply in pairs and add the results.
$\class{inputBox step25}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c0a0}}\hspace{40px}}~} \times \class{inputBox step25}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c0b0}}\hspace{40px}}~} + \class{inputBox step25}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c0a1}}\hspace{40px}}~} \times \class{inputBox step25}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c0b1}}\hspace{40px}}~} + \class{inputBox step25}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c0a2}}\hspace{40px}}~} \times \class{inputBox step25}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c0b2}}\hspace{40px}}~} + \class{inputBox step25}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c0a3}}\hspace{40px}}~} \times \class{inputBox step25}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c0b3}}\hspace{40px}}~}$ $[-5] \times [5] + [-1] \times [-8] + [-12] \times [4] + [4] \times [0]$
$= \class{inputBox step26}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c0p0}}\hspace{54px}}~} + \class{inputBox step26}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c0p1}}\hspace{54px}}~} + \class{inputBox step26}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c0p2}}\hspace{54px}}~} + \class{inputBox step26}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c0p3}}\hspace{54px}}~} = \class{inputBox step26}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c0R}}\hspace{54px}}~}$ $= [-25] + [8] + [-48] + [0] = [-65]$
Row 4, column 2: Hint: Row 4 of $A$ is $\begin{pmatrix} -5 & -1 & -12 & 4 \end{pmatrix}$. Column 2 of $B$ is $\begin{pmatrix} -9 \\ 3 \\ -7 \\ -8 \end{pmatrix}$. Multiply in pairs and add the results.
$\class{inputBox step27}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c1a0}}\hspace{40px}}~} \times \class{inputBox step27}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c1b0}}\hspace{40px}}~} + \class{inputBox step27}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c1a1}}\hspace{40px}}~} \times \class{inputBox step27}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c1b1}}\hspace{40px}}~} + \class{inputBox step27}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c1a2}}\hspace{40px}}~} \times \class{inputBox step27}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c1b2}}\hspace{40px}}~} + \class{inputBox step27}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c1a3}}\hspace{40px}}~} \times \class{inputBox step27}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c1b3}}\hspace{40px}}~}$ $[-5] \times [-9] + [-1] \times [3] + [-12] \times [-7] + [4] \times [-8]$
$= \class{inputBox step28}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c1p0}}\hspace{54px}}~} + \class{inputBox step28}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c1p1}}\hspace{54px}}~} + \class{inputBox step28}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c1p2}}\hspace{54px}}~} + \class{inputBox step28}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c1p3}}\hspace{54px}}~} = \class{inputBox step28}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c1R}}\hspace{54px}}~}$ $= [45] + [-3] + [84] + [-32] = [94]$
Row 4, column 3: Hint: Row 4 of $A$ is $\begin{pmatrix} -5 & -1 & -12 & 4 \end{pmatrix}$. Column 3 of $B$ is $\begin{pmatrix} 0 \\ 8 \\ -10 \\ -11 \end{pmatrix}$. Multiply in pairs and add the results.
$\class{inputBox step29}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c2a0}}\hspace{40px}}~} \times \class{inputBox step29}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c2b0}}\hspace{40px}}~} + \class{inputBox step29}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c2a1}}\hspace{40px}}~} \times \class{inputBox step29}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c2b1}}\hspace{40px}}~} + \class{inputBox step29}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c2a2}}\hspace{40px}}~} \times \class{inputBox step29}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c2b2}}\hspace{40px}}~} + \class{inputBox step29}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c2a3}}\hspace{40px}}~} \times \class{inputBox step29}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c2b3}}\hspace{40px}}~}$ $[-5] \times [0] + [-1] \times [8] + [-12] \times [-10] + [4] \times [-11]$
$= \class{inputBox step30}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c2p0}}\hspace{54px}}~} + \class{inputBox step30}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c2p1}}\hspace{54px}}~} + \class{inputBox step30}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c2p2}}\hspace{54px}}~} + \class{inputBox step30}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c2p3}}\hspace{54px}}~} = \class{inputBox step30}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c2R}}\hspace{54px}}~}$ $= [0] + [-8] + [120] + [-44] = [68]$
Row 4, column 4: Hint: Row 4 of $A$ is $\begin{pmatrix} -5 & -1 & -12 & 4 \end{pmatrix}$. Column 4 of $B$ is $\begin{pmatrix} 10 \\ -11 \\ -12 \\ 5 \end{pmatrix}$. Multiply in pairs and add the results.
$\class{inputBox step31}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c3a0}}\hspace{40px}}~} \times \class{inputBox step31}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c3b0}}\hspace{40px}}~} + \class{inputBox step31}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c3a1}}\hspace{40px}}~} \times \class{inputBox step31}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c3b1}}\hspace{40px}}~} + \class{inputBox step31}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c3a2}}\hspace{40px}}~} \times \class{inputBox step31}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c3b2}}\hspace{40px}}~} + \class{inputBox step31}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c3a3}}\hspace{40px}}~} \times \class{inputBox step31}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c3b3}}\hspace{40px}}~}$ $[-5] \times [10] + [-1] \times [-11] + [-12] \times [-12] + [4] \times [5]$
$= \class{inputBox step32}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c3p0}}\hspace{54px}}~} + \class{inputBox step32}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c3p1}}\hspace{54px}}~} + \class{inputBox step32}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c3p2}}\hspace{54px}}~} + \class{inputBox step32}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c3p3}}\hspace{54px}}~} = \class{inputBox step32}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c3R}}\hspace{54px}}~}$ $= [-50] + [11] + [144] + [20] = [125]$
Therefore,
$AB = \begin{pmatrix} \class{inputBox step33}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Fr0c0}}\hspace{54px}}~} & \class{inputBox step33}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Fr0c1}}\hspace{54px}}~} & \class{inputBox step33}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Fr0c2}}\hspace{54px}}~} & \class{inputBox step33}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Fr0c3}}\hspace{54px}}~} \\ \class{inputBox step33}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Fr1c0}}\hspace{54px}}~} & \class{inputBox step33}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Fr1c1}}\hspace{54px}}~} & \class{inputBox step33}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Fr1c2}}\hspace{54px}}~} & \class{inputBox step33}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Fr1c3}}\hspace{54px}}~} \\ \class{inputBox step33}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Fr2c0}}\hspace{54px}}~} & \class{inputBox step33}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Fr2c1}}\hspace{54px}}~} & \class{inputBox step33}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Fr2c2}}\hspace{54px}}~} & \class{inputBox step33}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Fr2c3}}\hspace{54px}}~} \\ \class{inputBox step33}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Fr3c0}}\hspace{54px}}~} & \class{inputBox step33}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Fr3c1}}\hspace{54px}}~} & \class{inputBox step33}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Fr3c2}}\hspace{54px}}~} & \class{inputBox step33}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Fr3c3}}\hspace{54px}}~} \end{pmatrix}$ $AB = \begin{pmatrix} [23] & [-25] & [-107] & [-119] \\ [112] & [-189] & [-177] & [-16] \\ [71] & [-121] & [-214] & [137] \\ [-65] & [94] & [68] & [125] \end{pmatrix}$