Cofactor/Examples/Arbitrary Example 2

Example of Cofactor

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:

\(\ds A_{21}\) \(=\) \(\ds \paren {-1}^3 D_{21}\)
\(\ds \) \(=\) \(\ds \paren {-1}^3 \begin {vmatrix} a_{12} & a_{13} \\ a_{32} & a_{33} \end {vmatrix}\)
\(\ds \) \(=\) \(\ds -\paren {a_{12} a_{33} - a_{13} a_{32} }\)
\(\ds \) \(=\) \(\ds a_{13} a_{32} - a_{12} a_{33}\)


Sources

  • 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): cofactor
  • 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): cofactor
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