Summary of determinant definition

The definition of determinant is summarized as follows:

In mathematics, determinant is an expression generated by solving linear equations. What is the definition of determinant? The following is the definition of determinant that I arranged for you. Welcome to read!

Definition of determinant

At first glance, the determinant of matrix A has a strange definition:

Where s g n(σ) is the sign difference of the arrangement σ.

For smaller matrices, such as second-order and third-order matrices, the determinant is expressed as follows, and some are just like the product of the main diagonal (upper left to lower right) minus the product of the secondary diagonal (upper right to lower left) (see red line and blue line in the figure).

Second order: third order:. However, for matrices with larger orders, the determinant has n! Item, not in this table.

Determinant of two-dimensional vector group

A determinant is the area of a parallelogram composed of a vector.

Let p be a two-dimensional directed Euclidean space, the so-called Euclidean plane. The determinants of the two vectors x and x' are:

Through calculation, we can know that the determinant represents the directed area of the parallelogram composed of vectors x and x'. And has the following properties:

If and only if the determinant is zero and the two vectors are * * * lines (linear correlation), the parallelogram degenerates into a straight line. If the counterclockwise direction is positive, the directed area means that the parallelogram area is valid and only if the vectors x and x' are arranged counterclockwise (as shown in the figure). Determinant is a bilinear mapping.