Describe Minors And Cofactors


Minors and cofactors in matrices are two of the most important terms since they are crucial as it determines the matrix’s adjoint and the inverse. It is necessary to find the minors of that matrix and then the cofactors of that matrix to find the determinants of a large square matrix (like 4 × 4).


Minor is described as the determinant of an element in a matrix obtained by removing the row and column where in the element is located.


\(D=\left| \begin{matrix} {{a}_{11}} & {{a}_{12}} & {{a}_{13}} \\ {{a}_{21}} & {{a}_{22}} & {{a}_{23}} \\ {{a}_{31}} & {{a}_{32}} & {{a}_{33}} \\ \end{matrix} \right|,\)

minor of \({{a}_{12}}\) is denoted as \({{M}_{12}}=\left| \begin{matrix} {{a}_{21}} & {{a}_{23}} \\ {{a}_{31}} & {{a}_{33}} \\ \end{matrix} \right|\)


The cofactor is identified as the minor which has been signed. A = (-1)i+j M, where M is minor of aij, is the cofactor of an element aij, denoted by Aij.

Cofactor of an element \({{a}_{i\,j}}\) is related to its minor as \({{C}_{i\,j}}={{\left( -1 \right)}^{i+j}}{{M}_{i\,j}},\) where ‘i’ denotes the \({{i}^{th}}\) row and ‘j’ denotes the \({{j}^{th}}\) column to which the element \({{a}_{i\,j}}\) belongs.

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