Determinant
The determinant is a value (here the Result) associated with a square matrix (here of Numbers separated by spaces).
This program is done in JavaScript, and the determinant is computed by solving the system of linear equations with the Gaussian method of elimination. After diagonalization of the given matrix, its determinants are found using the diagonal elements.
Otherwise, the determinant of a number is a given number, and that is the determinant of the first order
.
The second-order
(quadratic) determinant is calculated as in the figure to the right:
from the product of its elements on the main diagonal is subtracted the product of the elements on the secondary diagonal.
The following figure, on the left, shows the Rule of Sarrus, which we often use but only to calculate the third-order
determinant.
The first two columns (three elements each) of the given determinant are added to the right and the triple products of the
main diagonals
thus obtained are added and the products of (also three) elements on the secondary diagonals are subtracted from that sum.
In general, the determinant of a matrix of arbitrary size can be defined by the Leibniz formula or the Laplace expansion. The Laplace expansion, named after Pierre-Simon Laplace, also called cofactor expansion
, is an expression for the determinant |A| of an m × m square matrix A that is a weighted sum of the determinants of m sub-matrices of A, each of size (m−1) × (m−1). The Laplace expansion is of theoretical interest as one of several ways to view the determinant, as well as of practical use in determinant computation.
