# Cofactors

A computer finds the determinant from the pivots when the square matrix is reduced to upper triangular form using Gaussian elimination. However, originally the determinant was defined through cofactor expansion, which is credited to the great French mathematician, astronomer, and physicist Pierre-Simon marquis de Laplace (1749--1827) who is best known for his investigations into the stability of the solar system. Probably because he did not hold strong political views and was not a member of the aristocracy, he escaped imprisonment and execution during the French Revolution. Laplace was president of the Board of Longitude, aided in the organization of the metric system, helped found the scientific Society of Arcueil, and was created a marquis. He served for six weeks as minister of the interior under Napoleon.

Therefore, the cofactor expansion is also called the Laplace expansion, which is an expression for the determinant
\( \det{\bf A} = |{\bf A}| \) of an n × n matrix **A** that is a weighted sum
of the determinants of n sub-matrices of **A**, each of size (n−1) × (n−1). The Laplace expansion has mostly
educational and theoretical interest as one of several ways to view the determinant, but not of practical use in determinant
computation.

For a given n × n matrix \( {\bf A} = \left[ a_{ij} \right] , \) the
the **minor** of the entry in the *i*-th row and *j*-th column (also called the (i,j) minor) is
the determinant of the submatrix formed by deleting the *i*-th row and *j*-th column. This number is often
denoted **A**_{i,j} or **M**_{i,j}.

The term minor is apparently due to the English mathematician James Sylvester (who used it in 1850 paper).

**of**

*(i,j)*-cofactor**A**is the number

*C*

_{ij}given by \( C_{ij} = (-1)^{i+j} \det{\bf A}_{i,j} . \) The n × n matrix of cofactors is called the

**adjugate**of

**A**

Theorem: (**Cofactor Expansion or Laplace Expansion**)
The determinant of an n × n matrix **A** can be computed by a cofactor expansion
across any row or down any column. The expansion across the *i*-th row using cofactors
\( C_{ij} = (-1)^{i+j} \det{\bf A}_{i,j} . \) is

*j*-th column is

*a*

_{ij}in the matrix, regardless of the sign of

*a*

_{ij}itself. The factor \( (-1)^{i+j} \) determine the following checkerboard pattern of signs:

*n*-dimensional real space, and it is called the

**standard basis**. Its dimension is

*n*.

*i*-th row and

*j*-th column. Then the set \( {\bf M}_{i,j} \ : \ 1 \le i \le m , \ 1 \le j \le n \) is a basis for the set of all such real matrices. Its dimension is

*mn*.

*n*. It has dimension

*n*+1. ■

**Theorem: ** If **A** is a triangular matrix, then its determinant
is a product of the entries on the main diagonal. ■