finding is an important determinant of the action not only for linear algebra: for example, in the economy with the help of this calculation are solved systems of linear equations with many unknowns, widely used in economic problems.
concept determinant
determinant or determinant of the matrix is called an amount equal to the volume of the parallelepiped constructed on its row vectors or columns.Calculate this value only for a square matrix in which the number of rows and columns of the same.If the members of the matrix - the number, then the number will be determinant.
computing determinants
Keep in mind that there are some rules that can greatly facilitate such calculations.
Since the determinant of the matrix consisting of one member, it is the only element.Calculate the determinant of the second order is not difficult, it is enough of the product to take away the members of the main diagonal elements of the product located on the secondary diagonal.
Calculating the determinant of order 3 is easiest to carry out according to the rule of the triangle.To do this, perform the following steps:
- find the product of three members of the matrix located on its glavnoydiagonali.
- multiplied by three members who are on the triangle, the base of which are parallel to the main diagonal.
- repeat the first and the second action to the secondary diagonal.
- find the sum of all previous calculations, the resulting values with the numbers obtained in the third paragraph, take with a minus sign.
to easily spend finding the determinant of order 4 and higher dimensions, you need to consider the properties possessed by all determinants:
- value of the determinant does not change after the matrix transposition.
- Swap two adjacent rows or columns leads to a change in the sign of the determinant.
- If the matrix has two equal rows or columns, or all of the elements of the column (line) is zero, then its determinant is zero.
- Multiplication matrix by any number leads to increase of its determinant in the same number of times.
Using the above properties makes it easy to carry out to find the determinant of any order.For example, using the method of reduction of order at which the expansion of the determinant of the elements of row (column) multiplied by the cofactor.
Another way that simplifies finding the determinant of the matrix, is to bring it to a triangular form, when all the elements under the main diagonal are zero.In this case, the determinant of the matrix is calculated as the product of the numbers positioned on this diagonal.
And finally, it should be noted that the calculation of determinants, although it consists of a seemingly simple mathematical calculations, however, requires considerable care and perseverance.
