We prove the two computational tools for the determinant: row/column operations and cofactor expansions.

Column Operation The properties, stated for columns, are

changed to A by B |
determinant changed by |

r[col i] + [col j] |
det = BdetA |

[col i] ↔ [col j] |
det = - BdetA |

d[col i], d ≠ 0 |
det = Bd detA |

For the first operation, we have

`det`(..., ` u`, ...,

=

=

For the second operation, the property is exactly the alternating property, which is one of the defining properties of `det`.

For the third operation, the property is the (scalar part of) multilinearity property, which is also built into the definition of `det`.

Row Operation This follows from the column operation case and the property `det A` =

Cofactor Expansion Along Column Let us consider the cofactor expansion along the first column of a 4 by 4 matrix

= [Aa_{1} a_{2} a_{3} a_{4}] = [ |
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} |

By the linearity in the first column, we have

`det A` =

=

The first of the four determinants is

det(e_{1}, a_{2}, a_{3}, a_{4}) = det[ |
1 | a_{12} |
a_{13} |
a_{14} |
] = det[ |
1 | 0 | 0 | 0 | ] = D(A_{11}), |

0 | a_{22} |
a_{23} |
a_{24} |
0 | a_{22} |
a_{23} |
a_{24} |
|||

0 | a_{32} |
a_{33} |
a_{34} |
0 | a_{32} |
a_{33} |
a_{34} |
|||

0 | a_{42} |
a_{43} |
a_{44} |
0 | a_{42} |
a_{43} |
a_{44} |

where we used the (just proved) column operations (-`a`_{12})[col 1] + [col 2], (-`a`_{13})[col 1] + [col 3], (-`a`_{14})[col 1] + [col 4] for the second equality, and

A_{11} = [ |
a_{22} |
a_{23} |
a_{24} |
]. |

a_{32} |
a_{33} |
a_{34} |
||

a_{42} |
a_{43} |
a_{44} |

Since `D`(`A`_{11}) is clearly multilinear and alternating in columns of `A`_{11}, by this result, `D`(`A`_{11}) = `d det A`

The other determinants can be computed by first using the (just proved) row operations to change to a determinant of the form `det`(`e`_{1}, `a`_{2}, `a`_{3}, `a`_{4}) and then use the computation above. For example,

det(e_{2}, a_{2}, a_{3}, a_{4}) = det[ |
0 | a_{12} |
a_{13} |
a_{14} |
] = - det[ |
1 | a_{22} |
a_{23} |
a_{24} |
] = - detA_{21} = C_{21}. |

1 | a_{22} |
a_{23} |
a_{24} |
0 | a_{12} |
a_{13} |
a_{14} |
|||

0 | a_{32} |
a_{33} |
a_{34} |
0 | a_{32} |
a_{33} |
a_{34} |
|||

0 | a_{42} |
a_{43} |
a_{44} |
0 | a_{42} |
a_{43} |
a_{44} |

At then end, we get

`det A` =

For the expansions along some other columns, we may first use the (just proved) column operations to move the specific column to the first column and then use the result just proved. Finally, the idea of the argument for the 4 by 4 case clearly works for the general case.

Cofactor Expansion Along Row This follows from the column expansion case and the property `det A` =