Cofactor/Examples/Arbitrary Example 3

Example of Cofactor

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:

\(\ds \map {\tilde D} {2, 3 \mid 2, 4}\)

\(=\)

\(\ds \paren {-1}^{2 + 3 + 2 + 4} \map D {1, 4 \mid 1, 3}\)

\(\ds \)

\(=\)

\(\ds \paren {-1}^{11} \begin {vmatrix} a_{11} & a_{13} \\ a_{41} & a_{43} \\ \end {vmatrix}\)

\(\ds \)

\(=\)

\(\ds -\paren {a_{11} a_{43} - a_{41} a_{13} }\)

\(\ds \)

\(=\)

\(\ds a_{41} a_{13} - a_{11} a_{43}\)