Formulaic Maths Problems

Let $A = \begin{pmatrix} -12 & 11 \\ 2 & 11 \\ -10 & -12 \end{pmatrix}$ and $B = \begin{pmatrix} 5 & 6 & 8 & 10 \\ 9 & 9 & 9 & 8 \end{pmatrix}$. Find $AB$.

Row 1, column 1: Hint: Row 1 of $A$ is $\begin{pmatrix} -12 & 11 \end{pmatrix}$. Column 1 of $B$ is $\begin{pmatrix} 5 \\ 9 \end{pmatrix}$. Multiply in pairs and add the results.

$\class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c0a0}}\hspace{40px}}~} \times \class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c0b0}}\hspace{40px}}~} + \class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c0a1}}\hspace{40px}}~} \times \class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c0b1}}\hspace{40px}}~}$ $[-12] \times [5] + [11] \times [9]$

$= \class{inputBox step2}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c0p0}}\hspace{54px}}~} + \class{inputBox step2}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c0p1}}\hspace{54px}}~} = \class{inputBox step2}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c0R}}\hspace{54px}}~}$ $= [-60] + [99] = [39]$

Row 1, column 2: Hint: Row 1 of $A$ is $\begin{pmatrix} -12 & 11 \end{pmatrix}$. Column 2 of $B$ is $\begin{pmatrix} 6 \\ 9 \end{pmatrix}$. Multiply in pairs and add the results.

$\class{inputBox step3}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c1a0}}\hspace{40px}}~} \times \class{inputBox step3}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c1b0}}\hspace{40px}}~} + \class{inputBox step3}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c1a1}}\hspace{40px}}~} \times \class{inputBox step3}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c1b1}}\hspace{40px}}~}$ $[-12] \times [6] + [11] \times [9]$

$= \class{inputBox step4}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c1p0}}\hspace{54px}}~} + \class{inputBox step4}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c1p1}}\hspace{54px}}~} = \class{inputBox step4}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c1R}}\hspace{54px}}~}$ $= [-72] + [99] = [27]$

Row 1, column 3: Hint: Row 1 of $A$ is $\begin{pmatrix} -12 & 11 \end{pmatrix}$. Column 3 of $B$ is $\begin{pmatrix} 8 \\ 9 \end{pmatrix}$. Multiply in pairs and add the results.

$\class{inputBox step5}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c2a0}}\hspace{40px}}~} \times \class{inputBox step5}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c2b0}}\hspace{40px}}~} + \class{inputBox step5}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c2a1}}\hspace{40px}}~} \times \class{inputBox step5}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c2b1}}\hspace{40px}}~}$ $[-12] \times [8] + [11] \times [9]$

$= \class{inputBox step6}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c2p0}}\hspace{54px}}~} + \class{inputBox step6}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c2p1}}\hspace{54px}}~} = \class{inputBox step6}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c2R}}\hspace{54px}}~}$ $= [-96] + [99] = [3]$

Row 1, column 4: Hint: Row 1 of $A$ is $\begin{pmatrix} -12 & 11 \end{pmatrix}$. Column 4 of $B$ is $\begin{pmatrix} 10 \\ 8 \end{pmatrix}$. Multiply in pairs and add the results.

$\class{inputBox step7}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c3a0}}\hspace{40px}}~} \times \class{inputBox step7}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c3b0}}\hspace{40px}}~} + \class{inputBox step7}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c3a1}}\hspace{40px}}~} \times \class{inputBox step7}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c3b1}}\hspace{40px}}~}$ $[-12] \times [10] + [11] \times [8]$

$= \class{inputBox step8}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c3p0}}\hspace{54px}}~} + \class{inputBox step8}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c3p1}}\hspace{54px}}~} = \class{inputBox step8}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c3R}}\hspace{54px}}~}$ $= [-120] + [88] = [-32]$

Row 2, column 1: Hint: Row 2 of $A$ is $\begin{pmatrix} 2 & 11 \end{pmatrix}$. Column 1 of $B$ is $\begin{pmatrix} 5 \\ 9 \end{pmatrix}$. Multiply in pairs and add the results.

$\class{inputBox step9}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c0a0}}\hspace{40px}}~} \times \class{inputBox step9}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c0b0}}\hspace{40px}}~} + \class{inputBox step9}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c0a1}}\hspace{40px}}~} \times \class{inputBox step9}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c0b1}}\hspace{40px}}~}$ $[2] \times [5] + [11] \times [9]$

$= \class{inputBox step10}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c0p0}}\hspace{54px}}~} + \class{inputBox step10}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c0p1}}\hspace{54px}}~} = \class{inputBox step10}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c0R}}\hspace{54px}}~}$ $= [10] + [99] = [109]$

Row 2, column 2: Hint: Row 2 of $A$ is $\begin{pmatrix} 2 & 11 \end{pmatrix}$. Column 2 of $B$ is $\begin{pmatrix} 6 \\ 9 \end{pmatrix}$. Multiply in pairs and add the results.

$\class{inputBox step11}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c1a0}}\hspace{40px}}~} \times \class{inputBox step11}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c1b0}}\hspace{40px}}~} + \class{inputBox step11}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c1a1}}\hspace{40px}}~} \times \class{inputBox step11}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c1b1}}\hspace{40px}}~}$ $[2] \times [6] + [11] \times [9]$

$= \class{inputBox step12}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c1p0}}\hspace{54px}}~} + \class{inputBox step12}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c1p1}}\hspace{54px}}~} = \class{inputBox step12}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c1R}}\hspace{54px}}~}$ $= [12] + [99] = [111]$

Row 2, column 3: Hint: Row 2 of $A$ is $\begin{pmatrix} 2 & 11 \end{pmatrix}$. Column 3 of $B$ is $\begin{pmatrix} 8 \\ 9 \end{pmatrix}$. Multiply in pairs and add the results.

$\class{inputBox step13}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c2a0}}\hspace{40px}}~} \times \class{inputBox step13}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c2b0}}\hspace{40px}}~} + \class{inputBox step13}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c2a1}}\hspace{40px}}~} \times \class{inputBox step13}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c2b1}}\hspace{40px}}~}$ $[2] \times [8] + [11] \times [9]$

$= \class{inputBox step14}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c2p0}}\hspace{54px}}~} + \class{inputBox step14}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c2p1}}\hspace{54px}}~} = \class{inputBox step14}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c2R}}\hspace{54px}}~}$ $= [16] + [99] = [115]$

Row 2, column 4: Hint: Row 2 of $A$ is $\begin{pmatrix} 2 & 11 \end{pmatrix}$. Column 4 of $B$ is $\begin{pmatrix} 10 \\ 8 \end{pmatrix}$. Multiply in pairs and add the results.

$\class{inputBox step15}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c3a0}}\hspace{40px}}~} \times \class{inputBox step15}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c3b0}}\hspace{40px}}~} + \class{inputBox step15}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c3a1}}\hspace{40px}}~} \times \class{inputBox step15}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c3b1}}\hspace{40px}}~}$ $[2] \times [10] + [11] \times [8]$

$= \class{inputBox step16}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c3p0}}\hspace{54px}}~} + \class{inputBox step16}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c3p1}}\hspace{54px}}~} = \class{inputBox step16}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c3R}}\hspace{54px}}~}$ $= [20] + [88] = [108]$

Row 3, column 1: Hint: Row 3 of $A$ is $\begin{pmatrix} -10 & -12 \end{pmatrix}$. Column 1 of $B$ is $\begin{pmatrix} 5 \\ 9 \end{pmatrix}$. Multiply in pairs and add the results.

$\class{inputBox step17}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c0a0}}\hspace{40px}}~} \times \class{inputBox step17}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c0b0}}\hspace{40px}}~} + \class{inputBox step17}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c0a1}}\hspace{40px}}~} \times \class{inputBox step17}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c0b1}}\hspace{40px}}~}$ $[-10] \times [5] + [-12] \times [9]$

$= \class{inputBox step18}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c0p0}}\hspace{54px}}~} + \class{inputBox step18}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c0p1}}\hspace{54px}}~} = \class{inputBox step18}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c0R}}\hspace{54px}}~}$ $= [-50] + [-108] = [-158]$

Row 3, column 2: Hint: Row 3 of $A$ is $\begin{pmatrix} -10 & -12 \end{pmatrix}$. Column 2 of $B$ is $\begin{pmatrix} 6 \\ 9 \end{pmatrix}$. Multiply in pairs and add the results.

$\class{inputBox step19}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c1a0}}\hspace{40px}}~} \times \class{inputBox step19}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c1b0}}\hspace{40px}}~} + \class{inputBox step19}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c1a1}}\hspace{40px}}~} \times \class{inputBox step19}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c1b1}}\hspace{40px}}~}$ $[-10] \times [6] + [-12] \times [9]$

$= \class{inputBox step20}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c1p0}}\hspace{54px}}~} + \class{inputBox step20}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c1p1}}\hspace{54px}}~} = \class{inputBox step20}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c1R}}\hspace{54px}}~}$ $= [-60] + [-108] = [-168]$

Row 3, column 3: Hint: Row 3 of $A$ is $\begin{pmatrix} -10 & -12 \end{pmatrix}$. Column 3 of $B$ is $\begin{pmatrix} 8 \\ 9 \end{pmatrix}$. Multiply in pairs and add the results.

$\class{inputBox step21}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c2a0}}\hspace{40px}}~} \times \class{inputBox step21}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c2b0}}\hspace{40px}}~} + \class{inputBox step21}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c2a1}}\hspace{40px}}~} \times \class{inputBox step21}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c2b1}}\hspace{40px}}~}$ $[-10] \times [8] + [-12] \times [9]$

$= \class{inputBox step22}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c2p0}}\hspace{54px}}~} + \class{inputBox step22}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c2p1}}\hspace{54px}}~} = \class{inputBox step22}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c2R}}\hspace{54px}}~}$ $= [-80] + [-108] = [-188]$

Row 3, column 4: Hint: Row 3 of $A$ is $\begin{pmatrix} -10 & -12 \end{pmatrix}$. Column 4 of $B$ is $\begin{pmatrix} 10 \\ 8 \end{pmatrix}$. Multiply in pairs and add the results.

$\class{inputBox step23}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c3a0}}\hspace{40px}}~} \times \class{inputBox step23}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c3b0}}\hspace{40px}}~} + \class{inputBox step23}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c3a1}}\hspace{40px}}~} \times \class{inputBox step23}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c3b1}}\hspace{40px}}~}$ $[-10] \times [10] + [-12] \times [8]$

$= \class{inputBox step24}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c3p0}}\hspace{54px}}~} + \class{inputBox step24}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c3p1}}\hspace{54px}}~} = \class{inputBox step24}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c3R}}\hspace{54px}}~}$ $= [-100] + [-96] = [-196]$

Therefore,

$AB = \begin{pmatrix} \class{inputBox step25}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Fr0c0}}\hspace{54px}}~} & \class{inputBox step25}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Fr0c1}}\hspace{54px}}~} & \class{inputBox step25}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Fr0c2}}\hspace{54px}}~} & \class{inputBox step25}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Fr0c3}}\hspace{54px}}~} \\ \class{inputBox step25}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Fr1c0}}\hspace{54px}}~} & \class{inputBox step25}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Fr1c1}}\hspace{54px}}~} & \class{inputBox step25}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Fr1c2}}\hspace{54px}}~} & \class{inputBox step25}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Fr1c3}}\hspace{54px}}~} \\ \class{inputBox step25}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Fr2c0}}\hspace{54px}}~} & \class{inputBox step25}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Fr2c1}}\hspace{54px}}~} & \class{inputBox step25}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Fr2c2}}\hspace{54px}}~} & \class{inputBox step25}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Fr2c3}}\hspace{54px}}~} \end{pmatrix}$ $AB = \begin{pmatrix} [39] & [27] & [3] & [-32] \\ [109] & [111] & [115] & [108] \\ [-158] & [-168] & [-188] & [-196] \end{pmatrix}$