Finding the determinant of a matrix is an important action not only for linear algebra: for example, in economics, using this calculation, systems of linear equations with many unknowns are solved, which are widely used in economic problems.
Determinant concept
The determinant, or determinant, of a matrix is a value equal to the volume of a parallelepiped built on its row or column vectors. This value can be calculated only for a square matrix, which has the same number of rows and columns. If the members of the matrix are numbers, then the determinant will also be a number.
Calculation of determinants
It should be remembered that there are several rules that can greatly facilitate such calculations.
So the determinant of a matrix consisting of one member is equal to its only element. It is not difficult to calculate the second-order determinant; for this, it is enough to subtract the product of the elements located on the secondary diagonal from the product of the members of the main diagonal.
Calculation of the 3rd order determinant is easiest to doaccording to the triangle rule. To do this, perform the following actions:
- Find the product of three members of the matrix located on its main one
- Multiply by three terms located on triangles whose bases are parallel to the main diagonal.
- Repeat the first and second action for the secondary diagonal.
- Find the sum of all the values obtained in the previous calculations, while the numbers obtained in the third paragraph are taken with a minus sign.
diagonals.
To easily find the determinant of a matrix of 4th order, as well as higher dimensions, it is necessary to consider the properties that all determinants have:
- The value of the determinant does not change after matrix transposition.
- Changing the positions of 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 elements of the column (row) are zero, then its determinant is equal to zero.
- Multiplying the numbers of a matrix by any number leads to an increase in its determinant by the same number of times.
Using the above properties helps to easily find the determinant of a matrix of any order. For example, using the order reduction method for this, in which the determinant is expanded by the elements of the row (column) multiplied by the algebraic complement.
Another way that makes finding the determinant much easier
matrix is to bring it to a triangular form, when all elements under the main diagonal are equal to zero. In this case, the matrix determinant is calculated as the product of the numbers located on this diagonal.
And finally, I would like to note that the calculation of determinants, although it consists of seemingly simple mathematical calculations, however, requires considerable care and perseverance.
