Formulaic Maths Problems

Find the determinant of the matrix $A = \begin{pmatrix} 8 & -1 & -7 \\ 7 & 4 & -4 \\ 9 & -8 & -10 \end{pmatrix}$ by expansion of cofactors.

Expand. Hint: Multiply each element from the first row by its minor. The minor is the determinant of the matrix found by deleting that row and column.

$\operatorname{det} A = \class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Aa}}\hspace{35px}}~} \operatorname{det} \begin{pmatrix} \class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Ab11}}\hspace{35px}}~} & \class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Ab12}}\hspace{35px}}~} \\ \class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Ab21}}\hspace{35px}}~} & \class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Ab22}}\hspace{35px}}~} \end{pmatrix} - \class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Ac}}\hspace{35px}}~} \operatorname{det} \begin{pmatrix} \class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Ad11}}\hspace{35px}}~} & \class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Ad12}}\hspace{35px}}~} \\ \class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Ad21}}\hspace{35px}}~} & \class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Ad22}}\hspace{35px}}~} \end{pmatrix} + \class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Ae}}\hspace{35px}}~} \operatorname{det} \begin{pmatrix} \class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Af11}}\hspace{35px}}~} & \class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Af12}}\hspace{35px}}~} \\ \class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Af21}}\hspace{35px}}~} & \class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Af22}}\hspace{35px}}~} \end{pmatrix}$ $\operatorname{det} A = [8] \operatorname{det} \begin{pmatrix} [4] & [-4] \\ [-8] & [-10] \end{pmatrix} - [-1] \operatorname{det} \begin{pmatrix} [7] & [-4] \\ [9] & [-10] \end{pmatrix} + [-7] \operatorname{det} \begin{pmatrix} [7] & [4] \\ [9] & [-8] \end{pmatrix}$

Calculate the first minor. Hint: The determinant of a $2{\times}2$ matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ is $ad - bc$.

$\operatorname{det} \begin{pmatrix} 4 & -4 \\ -8 & -10 \end{pmatrix}= \class{inputBox step2}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{B1}}\hspace{54px}}~} \times \class{inputBox step2}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{B2}}\hspace{54px}}~} - \class{inputBox step2}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{B3}}\hspace{54px}}~} \times \class{inputBox step2}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{B4}}\hspace{54px}}~}$ $\operatorname{det} \begin{pmatrix} 4 & -4 \\ -8 & -10 \end{pmatrix}= [4] \times [-10] - [-4] \times [-8]$

$= \class{inputBox step3}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{C1}}\hspace{54px}}~} - \class{inputBox step3}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{C2}}\hspace{54px}}~} = \class{inputBox step3}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{CR}}\hspace{54px}}~}$ $= [-40] - [32] = [-72]$

Calculate the second minor.

$\operatorname{det} \begin{pmatrix} 7 & -4 \\ 9 & -10 \end{pmatrix}= \class{inputBox step4}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{D1}}\hspace{54px}}~} \times \class{inputBox step4}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{D2}}\hspace{54px}}~} - \class{inputBox step4}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{D3}}\hspace{54px}}~} \times \class{inputBox step4}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{D4}}\hspace{54px}}~}$ $\operatorname{det} \begin{pmatrix} 7 & -4 \\ 9 & -10 \end{pmatrix}= [7] \times [-10] - [-4] \times [9]$

$= \class{inputBox step5}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{E1}}\hspace{54px}}~} - \class{inputBox step5}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{E2}}\hspace{54px}}~} = \class{inputBox step5}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{ER}}\hspace{54px}}~}$ $= [-70] - [-36] = [-34]$

Calculate the third minor.

$\operatorname{det} \begin{pmatrix} 7 & 4 \\ 9 & -8 \end{pmatrix}= \class{inputBox step6}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{F1}}\hspace{54px}}~} \times \class{inputBox step6}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{F2}}\hspace{54px}}~} - \class{inputBox step6}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{F3}}\hspace{54px}}~} \times \class{inputBox step6}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{F4}}\hspace{54px}}~}$ $\operatorname{det} \begin{pmatrix} 7 & 4 \\ 9 & -8 \end{pmatrix}= [7] \times [-8] - [4] \times [9]$

$= \class{inputBox step7}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{G1}}\hspace{54px}}~} - \class{inputBox step7}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{G2}}\hspace{54px}}~} = \class{inputBox step7}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{GR}}\hspace{54px}}~}$ $= [-56] - [36] = [-92]$

Therefore, Hint: Substitute the minors into the first step.

$\operatorname{det} A = \class{inputBox step8}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{H1a}}\hspace{54px}}~} \times \class{inputBox step8}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{H1b}}\hspace{54px}}~} - \class{inputBox step8}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{H2a}}\hspace{54px}}~} \times \class{inputBox step8}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{H2b}}\hspace{54px}}~} + \class{inputBox step8}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{H3a}}\hspace{54px}}~} \times \class{inputBox step8}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{H3b}}\hspace{54px}}~}$ $\operatorname{det} A = [8] \times [-72] - [-1] \times [-34] + [-7] \times [-92]$

$= \class{inputBox step9}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{J1}}\hspace{54px}}~} - \class{inputBox step9}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{J2}}\hspace{54px}}~} + \class{inputBox step9}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{J3}}\hspace{54px}}~} = \class{inputBox step9}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{JR}}\hspace{54px}}~}$ $= [-576] - [34] + [644] = [34]$