Determinants

Minors and Cofactors


  • Minor of determinant: The minor of the element aij is a determinant of matrix obtained by detecting ith row and jth column in the determinant and it is denoted by Mij
    Ex:- \tt A = \begin{bmatrix}a & b \\c & d \end{bmatrix}
    Minor of a = d = M11
    Minor of b = c = M12
    Minor of c = b = M21
    Minor of d = a = M22
  • Cofactors: The cofactor of the element aij is denoted by cij and it is defined as
    cij = (−1)i + j Mij
    Ex: \tt A = \begin{bmatrix}a & b \\c & d \end{bmatrix}
    C11 = + d
    C12 = − c
    C21 = − b
    C22 = + a

View the Topic in this video From 29:06 To 34:30

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  • Minor of determinant: The minor of the element aij is a determinant of matrix obtained by detecting ith row and jth column in the determinant and it is denoted by Mij
  • Minors of 3 × 3 determinant:
    \tt A = \begin{bmatrix}a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{bmatrix}
    \tt M_{11} = \begin{vmatrix}b_{2} & c_{2} \\ b_{3} & c_{3} \end{vmatrix}
    \tt M_{12} = \begin{vmatrix}a_{2} & c_{2} \\ a_{3} & c_{3} \end{vmatrix}
    \tt M_{13} = \begin{vmatrix}a_{2} & b_{2} \\ a_{3} & b_{3} \end{vmatrix}
    \tt M_{21} = \begin{vmatrix}b_{1} & c_{1} \\ b_{3} & c_{3} \end{vmatrix}
    \tt M_{22} = \begin{vmatrix}a_{1} & c_{1} \\ a_{3} & c_{3} \end{vmatrix}
    \tt M_{23} = \begin{vmatrix}a_{1} & b_{1} \\ a_{3} & b_{3} \end{vmatrix}
    \tt M_{31} = \begin{vmatrix}b_{1} & c_{1} \\ b_{2} & c_{2} \end{vmatrix} 
    \tt M_{32} = \begin{vmatrix}a_{1} & c_{1} \\ a_{2} & c_{2} \end{vmatrix}
    \tt M_{33} = \begin{vmatrix}a_{1} & b_{1} \\ a_{2} & b_{2} \end{vmatrix}
  • Cofactors: The cofactor of the element aij is denoted by cij and it is defined as
    cij = (−1)i + j Mij
  • Cofactors of 3 × 3 order determinant:
    If A = \begin{bmatrix}a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{bmatrix} then the cofactors are
    C_{11} = +\begin{vmatrix}b_{2} & c_{2} \\ b_{3} & c_{3} \end{vmatrix}
    C_{12} = -\begin{vmatrix}a_{2} & c_{2} \\ a_{3} & c_{3} \end{vmatrix}
    C_{13} = +\begin{vmatrix}a_{2} & b_{2} \\ a_{3} & b_{3} \end{vmatrix}
    C_{21} = -\begin{vmatrix}b_{1} & c_{1} \\ b_{3} & c_{3} \end{vmatrix}
    C_{22} = +\begin{vmatrix}a_{1} & c_{1} \\ a_{3} & c_{3} \end{vmatrix}
    C_{23} = -\begin{vmatrix}a_{1} & b_{1} \\ a_{3} & b_{3} \end{vmatrix}
    C_{31} = +\begin{vmatrix}b_{1} & c_{1} \\ b_{2} & c_{2} \end{vmatrix}
    C_{32} = -\begin{vmatrix}a_{1} & c_{1} \\ a_{2} & c_{2} \end{vmatrix}
    C_{33} = +\begin{vmatrix}a_{1} & b_{1} \\ a_{2} & b_{2} \end{vmatrix}