# 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

Disclaimer: Compete.etutor.co may from time to time provide links to third party Internet sites under their respective fair use policy and it may from time to time provide materials from such third parties on this website. These third party sites and any third party materials are provided for viewers convenience and for non-commercial educational purpose only. Compete does not operate or control in any respect any information, products or services available on these third party sites. Compete.etutor.co makes no representations whatsoever concerning the content of these sites and the fact that compete.etutor.co has provided a link to such sites is NOT an endorsement, authorization, sponsorship, or affiliation by compete.etutor.co with respect to such sites, its services, the products displayed, its owners, or its providers.

• 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}