Understanding of the definition of an nth-order determinant

An nth-order determinant embodies the properties of an n*n square matrix, and has many applications in practice, though many of them are not very accessible if the basics are not sufficient.

The definition of a third-order determinant is

|a11 a12 a13|

|a21 a22 a23|

|a31 a32 a33|

= a11*a22*a33 + a12*a23*a31 + a21*a32*a13 - a31*a22*a13 - a21*a12*a33 - a32*a33 - a31*a22*a13 - a31*a22*a13 a12*a33 - a32*a23*a11

Nth order determinant can be defined by induction. Define the first order determinant |a| = a, let the (n-1)-order determinant has been defined before, then the n-order determinant can be defined by the formula that the determinant is expanded by the first row. Of course there are some other ways to define. They are all longer to write, so I won't write them here.

The most common application is based on Kramer's law to solve the n-dimensional system of equations with determinants, but this method is really a stupid way to solve the system of equations, in most cases not as simple as adding and subtracting the elimination method. For example, for a system of binary equations

a1*x + b1*y = c1

a2*x + b2*y = c2

the solution is

x = D1/D

y = D2/D

where

D =

|a1 b1|

|a2 b2|

D1 =

|c1 b1|

|c2 b2|

D2 =

|a1 c1|

|a2 c2|

Discretes can also be used to find the rank of a square matrix, the inverse of a square matrix, and so on, all of which are fundamental to linear algebra. The determinant itself is a concept of linear algebra.

Determinants are also commonly used in analytic geometry, for example, the area of a triangle in the plane is a third-order determinant:

|x1 y1 1|

|x2 y2 1| ÷ 2

|x3 y3 1|

Where (xi, yi, zi) is the three vertex coordinates.

Physically determinants are also commonly used to simplify some formulas. Engineering determinants are also powerful analytical tools.

The determinant is also used in engineering as a powerful analytical tool.