For the determinant of order 3 or above, it is generally converted into a triangular matrix by elementary row transformation, and then multiplied by the main diagonal elements.

In mathematics, determinant is a function of matrix A whose domain is det, and its value is scalar, which is denoted as det(A) or | |A |. Whether in linear algebra, polynomial theory or calculus (such as substitution integral method), determinant, as a basic mathematical tool, has important applications.

Determinant can be regarded as a generalization of the concept of directed area or volume in general Euclidean space. In other words, in N-dimensional Euclidean space, determinant describes the influence of a linear transformation on "volume".

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① A row (or a column) in determinant A is multiplied by the same number k, and the result is equal to kA.

② determinant a is equal to its transposed determinant at (the I-th row of at is the I-th column of a).

③ If there is a row (or a column) in the determinant of order n | α ij |; The determinant | α ij | is the sum of two determinants, where the first row (or column) is b 1, b2, ..., bn; The other is с 1, с2, …, с n; The elements in other rows (or columns) are exactly the same as those in | α ij |.

④ Two rows (or columns) in determinant A are interchanged, and the result is equal to -a..⑤ Multiply each element in one row (or column) of determinant A by a number, and then add it to each corresponding element in another row (or column), and the result is still A. ..