a_ 11 A_ 21 + a_ 12 A_ 22 + a_ 13 A_ 23 = 0 - Vidyadip

Question

$a_{11}A_{21} + a_{12}A_{22} + a_{13}A_{23} = 0$

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Answer

$\mathrm{A}=\left[\mathrm{a}_{\mathrm{ij}}\right]_{3 \times 3}=\left[\begin{array}{lll}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{array}\right]$(i) $\mathrm{A}_{21}=(-1)^{2+1} \mathrm{M}_{21}=-\left|\begin{array}{ll}a_{12} & a_{13} \\ a_{32} & a_{33}\end{array}\right|$
$= -(a_{12}a_{33} – a_{13}a_{32})$
$= -a_{12}a_{33} + a_{13}a_{32}$
$\mathrm{A}_{22}=(-1)^{2+2} \mathrm{M}_{22}=\left|\begin{array}{ll}a_{11} & a_{13} \\ a_{31} & a_{33}\end{array}\right|$
= a11a33 – a13a31
$\mathrm{A}_{23}=(-1)^{2+3} \mathrm{M}_{23}=-\left|\begin{array}{ll}a_{11} & a_{12} \\ a_{31} & a_{32}\end{array}\right|$
$= -(a11a32 – a12a31)$
$= -a11a32+ a12a31$
$\therefore a11A21 + a12A22 + a13A23$
$= a11(-a1233 + a13a32) + a12(a11a33 – a13a31) + a13(-a11a32 + a12a31)$
$= -a11a12a33 + a11a13a32 + a11a12a33 – a12a13a31 – a11a13a32 + a12a13a31$
$= 0$

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