a_ 11 A_ 11 + a_ 12 A_ 12 + a_ 13 A_ 13 = |A|

Question

$a_{11}A_{11} + a_{12}A_{12} + a_{13}A_{13} = |A|$

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Answer

$\begin{array}{l}\mathrm{A}_{11}=(-1)^{1+1} \mathrm{M}_{11}=\left|\begin{array}{ll}a_{22} & a_{23} \\ a_{32} & a_{33}\end{array}\right|\end{array} $
$ \mathrm{A}_{12}=(-1)^{1+2} \mathrm{M}_{12}=-\left|\begin{array}{ll}a_{21} & a_{23} \\ a_{31} & a_{33}\end{array}\right| $
$ \mathrm{A}_{13}=(-1)^{1+3} \mathrm{M}_{13}=\left|\begin{array}{ll}a_{21} & a_{22} \\ a_{31} & a_{32}\end{array}\right|$
$\begin{array}{l}\therefore a_{11} \mathrm{~A}_{11}+a_{12} \mathrm{~A}_{12}+a_{13} \mathrm{~A}_{13} \\ =a_{11}\left|\begin{array}{ll}a_{22} & a_{23} \\ a_{32} & a_{33}\end{array}\right|-a_{12}\left|\begin{array}{ll}a_{21} & a_{23} \\ a_{31} & a_{33}\end{array}\right|+a_{13}\left|\begin{array}{ll}a_{21} & a_{22} \\ a_{31} & a_{32}\end{array}\right|\end{array}$
$=\left|\begin{array}{lll}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{array}\right|=|\mathrm{A}|$.

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