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}