Find minors and cofactors of the elements a_ 11 , a_ 21 in the de… - Vidyadip

Question

Find minors and cofactors of the elements $a_{11}, a_{21}$ in the determinant $\Delta=\left|\begin{array}{lll} {a_{11}} & {a_{12}} & {a_{13}} \\ {a_{21}} & {a_{22}} & {a_{23}} \\ {a_{31}} & {a_{32}} & {a_{33}} \end{array}\right|$

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Answer

By definition of minors and cofactors, we have
Minor of $a_{11} = M_{11} = \left|\begin{array}{ll} {a_{22}} & {a_{23}} \\ {a_{32}} & {a_{33}} \end{array}\right| = a_{22}\ a_{33} – a_{23}\ a_{32}$
Cofactor of $a_{11} = A_{11} = (–1)^{1+1} M_{11} = a_{22}\ a_{33} – a_{23}\ a_{32}$
Minor of $a_{21} = M_{21} = \left|\begin{array}{ll} {a_{12}} & {a_{13}} \\ {a_{32}} & {a_{33}} \end{array}\right| = a_{12} a_{33} – a_{13} a_{32}$
Cofactor of $a_{21} = A_{21} = (–1)^{2+1}\ M_{21} = (–1) (a_{12}\ a_{33} – a_{13}\ a_{32}) = – a_{12}\ a_{33} + a_{13}\ a_{32}$

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