Cofactor/Examples
Examples of Cofactors
Arbitrary Example 1
Let $D$ be the determinant defined as:
- $D = \begin {vmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end {vmatrix}$
Then the cofactor of $2$ is defined as:
| \(\ds D_{12}\) | \(=\) | \(\ds \begin {vmatrix} 4 & 6 \\ 7 & 9 \end {vmatrix}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {-1}^{2 + 1} \paren {4 \times 9 - 6 \times 7}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 6\) |
Arbitrary Example 2
Let $D$ be the determinant defined as:
$\quad D = \begin {vmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end {vmatrix}$
Then the cofactor of $a_{2 1}$ is defined as:
| \(\ds A_{21}\) | \(=\) | \(\ds \paren {-1}^3 D_{21}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {-1}^3 \begin {vmatrix} a_{12} & a_{13} \\ a_{32} & a_{33} \end {vmatrix}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds -\paren {a_{12} a_{33} - a_{13} a_{32} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds a_{13} a_{32} - a_{12} a_{33}\) |
Arbitrary Example 3
Let $D$ be the determinant defined as:
$\quad D = \begin{vmatrix} a_{11} & a_{12} & a_{13} & a_{14} \\ a_{21} & a_{22} & a_{23} & a_{24} \\ a_{31} & a_{32} & a_{33} & a_{34} \\ a_{41} & a_{42} & a_{43} & a_{44} \\ \end{vmatrix}$
Let $\map D {2, 3 \mid 2, 4}$ be an order-$2$ minor of $D$.
Then the cofactor of $\map D {2, 3 \mid 2, 4}$ is given by:
| \(\ds \map {\tilde D} {2, 3 \mid 2, 4}\) | \(=\) | \(\ds \paren {-1}^{2 + 3 + 2 + 4} \map D {1, 4 \mid 1, 3}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {-1}^{11} \begin {vmatrix} a_{11} & a_{13} \\ a_{41} & a_{43} \\ \end {vmatrix}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds -\paren {a_{11} a_{43} - a_{41} a_{13} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds a_{41} a_{13} - a_{11} a_{43}\) |
Arbitrary Example 4
Let $D$ be the determinant defined as:
- $D = \begin {vmatrix} a & b & c \\ d & e & f \\ g & h & i \end {vmatrix}$
The cofactor of $e$ is defined as:
| \(\ds A_e\) | \(=\) | \(\ds A_{22}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {-1}^4 D_{22}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \begin {vmatrix} a & c \\ g & i \end {vmatrix}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {a i - g c}\) |
The cofactor of $d$ is defined as:
| \(\ds A_d\) | \(=\) | \(\ds A_{12}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {-1}^3 D_{12}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds -\begin {vmatrix} b & c \\ h & i \end {vmatrix}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {c h - b i}\) |