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Determinant

The determinant is a value (Result) associated with a square matrix (of Numbers separated by spaces).

Numbers :
Result :

Determinant 2x2

The determinant of a 2×2 matrix is defined by the graph on the right. In other words, to take the determinant of a 2×2 matrix, you multiply the top-left-to-bottom-right diagonal, and from this you subtract the product of bottom-left-to-top-right diagonal.

The next graph was taken from Wikipedia.

Sarrus rule

Sarrus' rule or Sarrus' scheme, on the left, is a method and a memorization scheme to compute the determinant of a 3×3 matrix. It is named after the French mathematician Pierre Frédéric Sarrus.

Write out the first 2 columns of the matrix to the right of the 3rd column, so that you have 5 columns in a row. Then add the products of the diagonals going from top to bottom (solid) and subtract the products of the diagonals going from bottom to top (dashed).

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 n × n square 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 is of theoretical interest as one of several ways to view the determinant, as well as of practical use in determinant computation.

On the next example, the determinant 4×4 is expanded along its third column.

Laplace expansion