## Determinants

# Minors and Cofactors

**Minor of determinant:**The minor of the element a_{ij}is a determinant of matrix obtained by detecting i^{th}row and j^{th}column in the determinant and it is denoted by M_{ij}

Ex:- \tt A = \begin{bmatrix}a & b \\c & d \end{bmatrix}

Minor of a = d = M_{11}

Minor of b = c = M_{12}

Minor of c = b = M_{21}

Minor of d = a = M_{22}**Cofactors:**The cofactor of the element a_{ij}is denoted by c_{ij}and it is defined as

c_{ij}= (−1)^{i + j}M_{ij}

Ex: \tt A = \begin{bmatrix}a & b \\c & d \end{bmatrix}

C_{11}= + d

C_{12}= − c

C_{21}= − b

C_{22}= + 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 a_{ij}is a determinant of matrix obtained by detecting i^{th}row and j^{th}column in the determinant and it is denoted by M_{ij}**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 a_{ij}is denoted by c_{ij}and it is defined as

c_{ij}= (−1)^{i + j}M_{ij}**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}