Editing Minor (linear algebra) (section)

===Cofactor expansion of the determinant===
{{main|Laplace expansion}}

The cofactors feature prominently in [[Laplace expansion|Laplace's formula]] for the expansion of determinants, which is a method of computing larger determinants in terms of smaller ones. Given an {{math|''n'' × ''n''}} matrix {{math|1='''A''' = (''a{{sub|ij}}'')}}, the determinant of {{math|'''A'''}}, denoted {{math|det('''A''')}}, can be written as the sum of the cofactors of any row or column of the matrix multiplied by the entries that generated them. In other words, defining <math>C_{ij} = (-1)^{i+j} M_{ij}</math> then the cofactor expansion along the {{mvar|j}}-th column gives:

<math display=block>\begin{align}
  \det(\mathbf A) &= a_{1j}C_{1j} + a_{2j}C_{2j} + a_{3j}C_{3j} + \cdots + a_{nj}C_{nj} \\[2pt]
  &= \sum_{i=1}^{n} a_{ij} C_{ij} \\[2pt]
  &= \sum_{i=1}^{n} a_{ij}(-1)^{i+j} M_{ij}
\end{align}</math>

The cofactor expansion along the {{mvar|i}}-th row gives:

<math display=block>\begin{align}
  \det(\mathbf A) &= a_{i1}C_{i1} + a_{i2}C_{i2} + a_{i3}C_{i3} + \cdots + a_{in}C_{in} \\[2pt]
  &= \sum_{j=1}^{n} a_{ij} C_{ij} \\[2pt]
  &= \sum_{j=1}^{n} a_{ij} (-1)^{i+j} M_{ij}
\end{align}</math>