Formulaic Maths Problems - get another problem: level 1, level 2, level 3 - link to this problem
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Let $A = \begin{pmatrix} 6 & 7 & 2 & -7 \\ 8 & 3 & 4 & 6 \\ 6 & -12 & 9 & -1 \end{pmatrix}$ and $B = \begin{pmatrix} 3 & -8 & -6 \\ -8 & 6 & 6 \\ -12 & -11 & -12 \\ 1 & 0 & 1 \end{pmatrix}$. Find $AB$.
Row 1, column 1: Hint: Row 1 of $A$ is $\begin{pmatrix} 6 & 7 & 2 & -7 \end{pmatrix}$. Column 1 of $B$ is $\begin{pmatrix} 3 \\ -8 \\ -12 \\ 1 \end{pmatrix}$. Multiply in pairs and add the results.
$\class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c0a0}}\hspace{40px}}~} \times \class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c0b0}}\hspace{40px}}~} + \class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c0a1}}\hspace{40px}}~} \times \class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c0b1}}\hspace{40px}}~} + \class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c0a2}}\hspace{40px}}~} \times \class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c0b2}}\hspace{40px}}~} + \class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c0a3}}\hspace{40px}}~} \times \class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c0b3}}\hspace{40px}}~}$ $6 \times 3 + 7 \times -8 + 2 \times -12 + -7 \times 1$
$= \class{inputBox step2}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c0p0}}\hspace{54px}}~} + \class{inputBox step2}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c0p1}}\hspace{54px}}~} + \class{inputBox step2}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c0p2}}\hspace{54px}}~} + \class{inputBox step2}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c0p3}}\hspace{54px}}~} = \class{inputBox step2}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c0R}}\hspace{54px}}~}$ $= 18 + -56 + -24 + -7 = -69$
Row 1, column 2: Hint: Row 1 of $A$ is $\begin{pmatrix} 6 & 7 & 2 & -7 \end{pmatrix}$. Column 2 of $B$ is $\begin{pmatrix} -8 \\ 6 \\ -11 \\ 0 \end{pmatrix}$. Multiply in pairs and add the results.
$\class{inputBox step3}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c1a0}}\hspace{40px}}~} \times \class{inputBox step3}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c1b0}}\hspace{40px}}~} + \class{inputBox step3}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c1a1}}\hspace{40px}}~} \times \class{inputBox step3}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c1b1}}\hspace{40px}}~} + \class{inputBox step3}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c1a2}}\hspace{40px}}~} \times \class{inputBox step3}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c1b2}}\hspace{40px}}~} + \class{inputBox step3}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c1a3}}\hspace{40px}}~} \times \class{inputBox step3}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c1b3}}\hspace{40px}}~}$ $6 \times -8 + 7 \times 6 + 2 \times -11 + -7 \times 0$
$= \class{inputBox step4}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c1p0}}\hspace{54px}}~} + \class{inputBox step4}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c1p1}}\hspace{54px}}~} + \class{inputBox step4}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c1p2}}\hspace{54px}}~} + \class{inputBox step4}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c1p3}}\hspace{54px}}~} = \class{inputBox step4}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c1R}}\hspace{54px}}~}$ $= -48 + 42 + -22 + 0 = -28$
Row 1, column 3: Hint: Row 1 of $A$ is $\begin{pmatrix} 6 & 7 & 2 & -7 \end{pmatrix}$. Column 3 of $B$ is $\begin{pmatrix} -6 \\ 6 \\ -12 \\ 1 \end{pmatrix}$. Multiply in pairs and add the results.
$\class{inputBox step5}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c2a0}}\hspace{40px}}~} \times \class{inputBox step5}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c2b0}}\hspace{40px}}~} + \class{inputBox step5}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c2a1}}\hspace{40px}}~} \times \class{inputBox step5}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c2b1}}\hspace{40px}}~} + \class{inputBox step5}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c2a2}}\hspace{40px}}~} \times \class{inputBox step5}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c2b2}}\hspace{40px}}~} + \class{inputBox step5}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c2a3}}\hspace{40px}}~} \times \class{inputBox step5}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c2b3}}\hspace{40px}}~}$ $6 \times -6 + 7 \times 6 + 2 \times -12 + -7 \times 1$
$= \class{inputBox step6}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c2p0}}\hspace{54px}}~} + \class{inputBox step6}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c2p1}}\hspace{54px}}~} + \class{inputBox step6}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c2p2}}\hspace{54px}}~} + \class{inputBox step6}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c2p3}}\hspace{54px}}~} = \class{inputBox step6}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c2R}}\hspace{54px}}~}$ $= -36 + 42 + -24 + -7 = -25$
Row 2, column 1: Hint: Row 2 of $A$ is $\begin{pmatrix} 8 & 3 & 4 & 6 \end{pmatrix}$. Column 1 of $B$ is $\begin{pmatrix} 3 \\ -8 \\ -12 \\ 1 \end{pmatrix}$. Multiply in pairs and add the results.
$\class{inputBox step7}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c0a0}}\hspace{40px}}~} \times \class{inputBox step7}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c0b0}}\hspace{40px}}~} + \class{inputBox step7}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c0a1}}\hspace{40px}}~} \times \class{inputBox step7}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c0b1}}\hspace{40px}}~} + \class{inputBox step7}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c0a2}}\hspace{40px}}~} \times \class{inputBox step7}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c0b2}}\hspace{40px}}~} + \class{inputBox step7}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c0a3}}\hspace{40px}}~} \times \class{inputBox step7}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c0b3}}\hspace{40px}}~}$ $8 \times 3 + 3 \times -8 + 4 \times -12 + 6 \times 1$
$= \class{inputBox step8}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c0p0}}\hspace{54px}}~} + \class{inputBox step8}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c0p1}}\hspace{54px}}~} + \class{inputBox step8}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c0p2}}\hspace{54px}}~} + \class{inputBox step8}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c0p3}}\hspace{54px}}~} = \class{inputBox step8}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c0R}}\hspace{54px}}~}$ $= 24 + -24 + -48 + 6 = -42$
Row 2, column 2: Hint: Row 2 of $A$ is $\begin{pmatrix} 8 & 3 & 4 & 6 \end{pmatrix}$. Column 2 of $B$ is $\begin{pmatrix} -8 \\ 6 \\ -11 \\ 0 \end{pmatrix}$. Multiply in pairs and add the results.
$\class{inputBox step9}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c1a0}}\hspace{40px}}~} \times \class{inputBox step9}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c1b0}}\hspace{40px}}~} + \class{inputBox step9}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c1a1}}\hspace{40px}}~} \times \class{inputBox step9}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c1b1}}\hspace{40px}}~} + \class{inputBox step9}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c1a2}}\hspace{40px}}~} \times \class{inputBox step9}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c1b2}}\hspace{40px}}~} + \class{inputBox step9}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c1a3}}\hspace{40px}}~} \times \class{inputBox step9}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c1b3}}\hspace{40px}}~}$ $8 \times -8 + 3 \times 6 + 4 \times -11 + 6 \times 0$
$= \class{inputBox step10}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c1p0}}\hspace{54px}}~} + \class{inputBox step10}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c1p1}}\hspace{54px}}~} + \class{inputBox step10}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c1p2}}\hspace{54px}}~} + \class{inputBox step10}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c1p3}}\hspace{54px}}~} = \class{inputBox step10}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c1R}}\hspace{54px}}~}$ $= -64 + 18 + -44 + 0 = -90$
Row 2, column 3: Hint: Row 2 of $A$ is $\begin{pmatrix} 8 & 3 & 4 & 6 \end{pmatrix}$. Column 3 of $B$ is $\begin{pmatrix} -6 \\ 6 \\ -12 \\ 1 \end{pmatrix}$. Multiply in pairs and add the results.
$\class{inputBox step11}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c2a0}}\hspace{40px}}~} \times \class{inputBox step11}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c2b0}}\hspace{40px}}~} + \class{inputBox step11}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c2a1}}\hspace{40px}}~} \times \class{inputBox step11}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c2b1}}\hspace{40px}}~} + \class{inputBox step11}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c2a2}}\hspace{40px}}~} \times \class{inputBox step11}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c2b2}}\hspace{40px}}~} + \class{inputBox step11}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c2a3}}\hspace{40px}}~} \times \class{inputBox step11}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c2b3}}\hspace{40px}}~}$ $8 \times -6 + 3 \times 6 + 4 \times -12 + 6 \times 1$
$= \class{inputBox step12}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c2p0}}\hspace{54px}}~} + \class{inputBox step12}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c2p1}}\hspace{54px}}~} + \class{inputBox step12}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c2p2}}\hspace{54px}}~} + \class{inputBox step12}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c2p3}}\hspace{54px}}~} = \class{inputBox step12}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c2R}}\hspace{54px}}~}$ $= -48 + 18 + -48 + 6 = -72$
Row 3, column 1: Hint: Row 3 of $A$ is $\begin{pmatrix} 6 & -12 & 9 & -1 \end{pmatrix}$. Column 1 of $B$ is $\begin{pmatrix} 3 \\ -8 \\ -12 \\ 1 \end{pmatrix}$. Multiply in pairs and add the results.
$\class{inputBox step13}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c0a0}}\hspace{40px}}~} \times \class{inputBox step13}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c0b0}}\hspace{40px}}~} + \class{inputBox step13}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c0a1}}\hspace{40px}}~} \times \class{inputBox step13}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c0b1}}\hspace{40px}}~} + \class{inputBox step13}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c0a2}}\hspace{40px}}~} \times \class{inputBox step13}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c0b2}}\hspace{40px}}~} + \class{inputBox step13}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c0a3}}\hspace{40px}}~} \times \class{inputBox step13}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c0b3}}\hspace{40px}}~}$ $6 \times 3 + -12 \times -8 + 9 \times -12 + -1 \times 1$
$= \class{inputBox step14}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c0p0}}\hspace{54px}}~} + \class{inputBox step14}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c0p1}}\hspace{54px}}~} + \class{inputBox step14}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c0p2}}\hspace{54px}}~} + \class{inputBox step14}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c0p3}}\hspace{54px}}~} = \class{inputBox step14}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c0R}}\hspace{54px}}~}$ $= 18 + 96 + -108 + -1 = 5$
Row 3, column 2: Hint: Row 3 of $A$ is $\begin{pmatrix} 6 & -12 & 9 & -1 \end{pmatrix}$. Column 2 of $B$ is $\begin{pmatrix} -8 \\ 6 \\ -11 \\ 0 \end{pmatrix}$. Multiply in pairs and add the results.
$\class{inputBox step15}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c1a0}}\hspace{40px}}~} \times \class{inputBox step15}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c1b0}}\hspace{40px}}~} + \class{inputBox step15}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c1a1}}\hspace{40px}}~} \times \class{inputBox step15}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c1b1}}\hspace{40px}}~} + \class{inputBox step15}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c1a2}}\hspace{40px}}~} \times \class{inputBox step15}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c1b2}}\hspace{40px}}~} + \class{inputBox step15}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c1a3}}\hspace{40px}}~} \times \class{inputBox step15}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c1b3}}\hspace{40px}}~}$ $6 \times -8 + -12 \times 6 + 9 \times -11 + -1 \times 0$
$= \class{inputBox step16}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c1p0}}\hspace{54px}}~} + \class{inputBox step16}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c1p1}}\hspace{54px}}~} + \class{inputBox step16}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c1p2}}\hspace{54px}}~} + \class{inputBox step16}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c1p3}}\hspace{54px}}~} = \class{inputBox step16}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c1R}}\hspace{54px}}~}$ $= -48 + -72 + -99 + 0 = -219$
Row 3, column 3: Hint: Row 3 of $A$ is $\begin{pmatrix} 6 & -12 & 9 & -1 \end{pmatrix}$. Column 3 of $B$ is $\begin{pmatrix} -6 \\ 6 \\ -12 \\ 1 \end{pmatrix}$. Multiply in pairs and add the results.
$\class{inputBox step17}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c2a0}}\hspace{40px}}~} \times \class{inputBox step17}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c2b0}}\hspace{40px}}~} + \class{inputBox step17}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c2a1}}\hspace{40px}}~} \times \class{inputBox step17}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c2b1}}\hspace{40px}}~} + \class{inputBox step17}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c2a2}}\hspace{40px}}~} \times \class{inputBox step17}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c2b2}}\hspace{40px}}~} + \class{inputBox step17}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c2a3}}\hspace{40px}}~} \times \class{inputBox step17}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c2b3}}\hspace{40px}}~}$ $6 \times -6 + -12 \times 6 + 9 \times -12 + -1 \times 1$
$= \class{inputBox step18}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c2p0}}\hspace{54px}}~} + \class{inputBox step18}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c2p1}}\hspace{54px}}~} + \class{inputBox step18}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c2p2}}\hspace{54px}}~} + \class{inputBox step18}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c2p3}}\hspace{54px}}~} = \class{inputBox step18}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c2R}}\hspace{54px}}~}$ $= -36 + -72 + -108 + -1 = -217$
Therefore,
$AB = \begin{pmatrix} \class{inputBox step19}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Fr0c0}}\hspace{54px}}~} & \class{inputBox step19}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Fr0c1}}\hspace{54px}}~} & \class{inputBox step19}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Fr0c2}}\hspace{54px}}~} \\ \class{inputBox step19}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Fr1c0}}\hspace{54px}}~} & \class{inputBox step19}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Fr1c1}}\hspace{54px}}~} & \class{inputBox step19}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Fr1c2}}\hspace{54px}}~} \\ \class{inputBox step19}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Fr2c0}}\hspace{54px}}~} & \class{inputBox step19}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Fr2c1}}\hspace{54px}}~} & \class{inputBox step19}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Fr2c2}}\hspace{54px}}~} \end{pmatrix}$ $AB = \begin{pmatrix} -69 & -28 & -25 \\ -42 & -90 & -72 \\ 5 & -219 & -217 \end{pmatrix}$