It is very important to make full use of the characteristics of determinant to simplify determinant. According to the characteristics of determinant, the second-order reduction method makes a row (column) contain only one non-zero element, and then expands it according to the row (column). Once expanded, the determinant is reduced by one order, which is effective for low-order digital determinants. Divide the determinant by three and (product) simplifies a complex determinant into two simpler determinants. Fourthly, according to the characteristics of the determinant, the Vandermonde determinant is appropriately deformed (using the properties of the determinant, such as: extracting the common factor; Interchange two rows (columns); Multiply one row by an appropriate number and add it to another row (column); ..) to convert the required determinant into a known or simple form. Vandermonde determinant is one of them. This deformation method is the most commonly used method to calculate determinant. 5. Mathematical induction When sum is the determinant of the same type, you can consider using mathematical induction to find it. The recursive relation established by the six inverse method is pushed down step by step, and the value of is obtained. Sometimes, you can also use to find a recursive relationship and finally use the obtained value. The heptagon method requires: 1 to keep the original determinant unchanged; The value of the new determinant is easy to calculate. Select the added rows and columns according to the needs and the characteristics of the original determinant. Margin method is suitable for rows (columns) with the same letters, and can also be used when the elements in the columns (rows) are multiples of n- 1 elements respectively. There are many methods to calculate determinant by eight-synthesis method, which are also very flexible. The general principle is: make full use of the characteristics of determinant, use the properties of determinant and the above-mentioned common methods, and sometimes the value of determinant can be obtained more simply by comprehensive application of the above methods; Sometimes the value of determinant can be found in many ways. Definition of nine determinants