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Category : JEE Main & Advanced

(1) Minor of an element : If we take the element of the determinant and delete (remove) the row and column containing that element, the determinant left is called the minor of that element. It is denoted by \[{{M}_{ij}}\].

Consider the determinant \[\Delta =\left| \,\begin{matrix} {{a}_{11}} & {{a}_{12}} & {{a}_{13}}  \\ {{a}_{21}} & {{a}_{22}} & {{a}_{23}}  \\ {{a}_{31}} & {{a}_{32}} & {{a}_{33}}  \\ \end{matrix}\, \right|\],

then determinant of minors \[M=\left| \,\begin{matrix} {{M}_{11}} & {{M}_{12}} & {{M}_{13}}  \\ {{M}_{21}} & {{M}_{22}} & {{M}_{23}}  \\ {{M}_{31}} & {{M}_{32}} & {{M}_{33}}  \\ \end{matrix}\, \right|\]

where  \[{{M}_{11}}=\] minor of \[{{a}_{11}}=\left| \,\begin{matrix} {{a}_{22}} & {{a}_{23}}  \\ {{a}_{32}} & {{a}_{33}}  \\ \end{matrix}\, \right|\]  \[{{M}_{12}}=\]minor of  \[{{a}_{12}}=\left| \,\begin{matrix} {{a}_{21}} & {{a}_{23}}  \\ {{a}_{31}} & {{a}_{33}}  \\ \end{matrix}\, \right|\] \[{{M}_{13}}=\] minor of \[{{a}_{13}}=\left| \,\begin{matrix} {{a}_{21}} & {{a}_{22}}  \\  {{a}_{31}} & {{a}_{32}}  \\ \end{matrix}\, \right|\]

Similarly, we can find the minors of other elements . Using this concept the value of determinant can be

\[\Delta ={{a}_{11}}{{M}_{11}}-{{a}_{12}}{{M}_{12}}+{{a}_{13}}{{M}_{13}}\]

or, \[\Delta =-{{a}_{21}}{{M}_{21}}+{{a}_{22}}{{M}_{22}}-{{a}_{23}}{{M}_{23}}\]

or,  \[\Delta ={{a}_{31}}{{M}_{31}}-{{a}_{32}}{{M}_{32}}+{{a}_{33}}{{M}_{33}}\].

(2) Cofactor of an element : The cofactor of an element \[{{a}_{ij}}\] (i.e. the element in the \[{{i}^{th}}\] row and \[{{j}^{th}}\] column) is defined as \[{{(-1)}^{i+j}}\] times the minor of that element. It is denoted by \[{{C}_{ij}}\] or \[{{A}_{ij}}\] or \[{{F}_{ij}}\]. \[{{C}_{ij}}={{(-1)}^{i+j}}{{M}_{ij}}\]

If \[\Delta =\left| \,\begin{matrix} {{a}_{11}} & {{a}_{12}} & {{a}_{13}}  \\  {{a}_{21}} & {{a}_{22}} & {{a}_{23}}  \\ {{a}_{31}} & {{a}_{32}} & {{a}_{33}}  \\ \end{matrix}\, \right|\], then determinant of cofactors is \[C=\left| \,\begin{matrix} {{C}_{11}} & {{C}_{12}} & {{C}_{13}}  \\ {{C}_{21}} & {{C}_{22}} & {{C}_{23}}  \\ {{C}_{31}} & {{C}_{32}} & {{C}_{33}}  \\ \end{matrix}\, \right|\]

where \[{{C}_{11}}={{(-1)}^{1+1}}{{M}_{11}}=+{{M}_{11}}\], \[{{C}_{12}}={{(-1)}^{1+2}}{{M}_{12}}=-{{M}_{12}}\]  and  \[{{C}_{13}}={{(-1)}^{1+3}}{{M}_{13}}=+{{M}_{13}}\]

Similarly, we can find the cofactors of other elements.