JEE Main & Advanced Mathematics Determinants & Matrices Minors and Cofactors

Minors and Cofactors

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.

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