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 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).
For a given n × n matrix \( {\bf A} = \left[ a_{ij} \right] , \) the (i,j)-cofactor
of A is the number C_{ij} given by \( C_{ij} = (-1)^{i+j} M_{i,j} = (-1)^{i+j} \det{\bf A}_{i,j} . \)
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
The plus or minus sign in the (i,j)-cofactor depends on the position of 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:
In the following Mathematica commands, we will compute the determinant
of A by expanding along the first row and then also the first and third
columns, using the Det and Drop commands:
Example: In \( \mathbb{R}^n , \) the vectors
\( e_1 [1,0,0,\ldots , 0] , \quad e_2 =[0,1,0,\ldots , 0], \quad \ldots , e_n =[0,0,\ldots , 0,1] \)
form a basis for n-dimensional real space, and it is called the standard basis. Its dimension is n.
Example: Let us consider the set of all real \( m \times n \)
matrices, and let \( {\bf M}_{i,j} \) denote the matrix whose only nonzero entry is a 1 in
the 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.
Example: The set of monomials \( \left\{ 1, x, x^2 , \ldots , x^n \right\} \)
form a basis in the set of all polynomials of degree up to n. It has dimension n+1.
■
Example: The infinite set of monomials \( \left\{ 1, x, x^2 , \ldots , x^n , \ldots \right\} \)
form a basis in the set of all polynomials.
■
Theorem:
If A is a triangular matrix, then its determinant
is a product of the entries on the main diagonal. ▣
Avera, V. and De Simone, A., An elementary proof of Laplace's formula on determinants, International Journal of Mathematical Education in Science and Technology, Volume 43, 2012 - Issue 3, https://doi.org/10.1080/0020739X.2011.592618