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 the economy, this system solves systems of linear equations with many unknowns that are widely used in economic problems.

The concept of a determinant

The determinant, or determinant, of a matrix is a quantity equal to the volume of a parallelepiped constructed on its row vectors or columns. Calculate this value 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 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 term is equal to its single element. To calculate the determinant of the second order is not difficult, for this it is sufficient to subtract the product of the elements located on the secondary diagonal from the product of the main diagonal members.

The calculation of the determinant of order 3 is most easily done by the rule of a triangle. To do this, we perform the following actions:

1. We find the product of three terms of the matrix located on its principal Diagonals.
2. We multiply three members on triangles whose bases are parallel to the main diagonal.
3. We repeat the first and second actions for an auxiliary diagonal.
4. We find the sum of all the values obtained in the previous calculations, while the numbers obtained in the third paragraph are taken with the minus sign.

In order to easily find the determinant of a matrix of order 4, as well as higher dimensions, it is necessary to consider the properties that all the 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 there are two equal rows or columns in the matrix, or all the elements of the column (rows) are zero, then its determinant is zero.
4. Multiplication of matrix numbers 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 for this purpose the method of decreasing the order at which the determinant is decomposed into elements of a row (column) multiplied by an algebraic complement.

Another way, which greatly simplifies finding the determinant Matrix, is its reduction to a triangular form, when all the elements 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, but requires considerable care and perseverance.