1, the basic concept of determinant
Determinant is a basic tool in linear algebra, and it is a rectangular table composed of n rows and n columns. The value of determinant is calculated by the algebraic cofactor of each element according to certain rules. Algebraic cofactor is an algebraic cofactor that multiplies the values of the determinant after removing one element from the determinant.
2. Algebraic cofactor expansion method
The expansion method of algebraic cofactor is a method to calculate the value of determinant by constructing the expansion of n-order determinant and using the properties of algebraic cofactor Based on the definition and properties of algebraic cofactors, this method expands the determinant into the sum of the products of several algebraic cofactors, thus simplifying the calculation process of determinant.
3. Recursive method
Recursive method is a method to recursively obtain the value of high-order determinant from the value of low-order determinant by using the properties and formulas of determinant. Based on the derivation of recursive formula, this method transforms high-order determinant into low-order determinant, thus reducing the calculation difficulty of determinant.
Application of determinant calculation in mathematics and physics
1, matrix operation
In matrix operation, the value of determinant can be used to judge whether the matrix is reversible. If the determinant value of a matrix is not zero, the matrix is reversible; Conversely, if the determinant value is zero, the matrix is irreversible. Determinants are also used for matrix multiplication, addition and subtraction.
2. Solve linear equations
When solving linear equations, the value of determinant can be used to solve equations. By applying Gauss elimination method or Cramer rule to the coefficient matrix of the equation, the equation can be solved by using the properties and formulas of determinant.
3, eigenvalue calculation
In eigenvalue calculation, the value of determinant can be used to calculate the eigenvalue and eigenvector of matrix. The square matrix is transformed into diagonal matrix by similarity transformation. According to the properties and formulas of diagonal matrix, the eigenvalues and eigenvectors of the matrix can be calculated.