How to find the determinant of a matrix?

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 widely used in economic problems.

The definition of a qualifier

The determinant, or determinant, of a matrix is ​​called the value equal to the volume of the parallelepiped built on its row vectors or columns. This value can be calculated only for a square matrix in which the number of rows and columns is the same. If the members of the matrix are numbers, then the determinant will 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 take the product of elements located on the side diagonal from the product of the members of the main diagonal.

The calculation of the determinant of 3 orders is easiest to carry out according to the triangle rule. To do this, perform the following steps:

1. We find the product of three members of the matrix located on its main
   
   diagonals.
2. We multiply three members located on the triangles, the bases of which are parallel to the main diagonal.
3. Repeat the first and second action for the side diagonal.
4. We find the sum of all the values ​​obtained in previous calculations, while taking the numbers obtained in the third paragraph with a minus sign.

In order to easily find the determinant of a 4-order matrix, as well as higher dimensions, it is necessary to consider the properties that all determinants possess:

1. The value of the determinant does not change after transposition of the matrix.
2. Rearrangement of two adjacent rows or columns leads to a change in the sign of the determinant.
3. If the matrix has two equal rows or columns, or all elements of the column (row) are zero, then its determinant is zero.
4. Multiplying the numbers of the 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 method of decreasing the order for which the determinant is decomposed into elements of a row (column) multiplied by an algebraic complement.

Another way that greatly simplifies finding the determinant

matrix, is to bring it to a triangular form, when all the elements located under the main diagonal are equal to zero. In this case, the determinant of the matrix 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 attention and perseverance.