The determinant of a matrix is a polynomial from all possible products of its elements.
One way to calculate the determinant is to expand the matrix by a column into additional minors (submatrices).
You will need
- - a pen
- - paper
It is known that the determinant of the matrix of the secondOrder is calculated as follows: the product of the elements of the main diagonal is subtracted from the product of the elements of the main diagonal. Therefore, it is convenient to decompose the matrix into second-order minors and then calculate the determinants of these minors, as well as the determinant of the original matrix.
The figure shows the formula for calculatingDeterminant of any matrix. Using it, we decompose the matrix first into third-order minors, and then each minor obtained in the second-order minors, which makes it easy to calculate the determinant of the matrices.
We decompose the original matrix byAdditional matrices of size 3 by 3. Additional matrices, or minors, are formed by deleting from the original matrix one row and one column. In a series of polynomials such minors enter multiplied by the element of the matrix to which they are complementary, the sign of the polynomial is determined by the power of -1, which is the sum of the indices of the element.
Now each of the matrices of the third orderWe decompose in the same way onto second-order matrices. We find the determinant of each such matrix and obtain a series of polynomials from the elements of the original matrix, then purely arithmetical calculations go on.