Practical application and finding the inverse matrix

A matrix is a table that is filled with a certain set of numbers in a certain order. This term was put into circulation by the eminent English theoretical theorist James Sylvester. He is one of the founders of the theory of applying these mathematical elements.

To date, they have found wide application in carrying out various calculations that are built on the basis of such a method as, for example, finding an inverse matrix in various branches of human activity. This method is based on the determination of unknown parameters of the system of various equations and is often used in the conduct of economic calculations.

There are the following particular cases of these mathematical components: lowercase, column, zero, square, diagonal, single. The lowercase consists of only one row of elements, and the column one consists of one column of numbers. Zero - all its elements are 0. In a square such a mathematical element, the number of columns is equal to the number of rows. In turn, in diagonal elements located on the main diagonal are different from "0", and the rest in it should be equal to "0". Single - this is one of the subspecies of the diagonal matrix. She has only "1" on the main diagonal.

Examples of matrices:

Where: A _k is a general notation, and _ij are elements,

(A) -2-th order;

(B) - lowercase;

(C) -3-th order;

(D) is an example of a unit table of the second order;

There is also an inverse matrix, the definition of which is as follows. When multiplying by the original inverse table, a single one is obtained. A lot of methods have been developed that ensure the finding of the inverse matrix. The simplest of them is based on the definition of algebraic complements and determinant (it is also sometimes called the determinant).

The determinant of the matrix is the expression a ₁₁ a ₂₂ -a ₁₂ a _{21, it is} denoted as | A |. The above formula is valid for the table corresponding to the second order. There are formulas for determinants of matrices of higher order. An obligatory condition for the existence of a determinant is that the table must be square. In practice, this element of this theory is most often used in such a procedure as finding an inverse matrix.

The second important component, by means of which it is possible to find the values of its elements, is an algebraic complement. It is calculated by the formula: A _ij = (- 1) ^{i + j} * M _ij , where M is a minor. Essentially, this is an additional determinant that can be obtained by mentally deleting the row and column in which the given element is located. For example, for a table corresponding to the second order, which was given earlier in the text, for an element a _{11, the} element a ₂₂ is an algebraic complement.

The inverse matrix is found in 3 stages. At the first stage, the determinant is determined. In the next step, all algebraic complements, which are then written in accordance with their indices, yield a table of algebraic complements. At the final stage, an inverse matrix is obtained, the finding of which terminates by multiplying each algebraic complement by the determinant.

Most often, matrices are used in economic calculations. With their help, you can easily and quickly process a large amount of information. In this case, the final result will be presented in a form convenient for perception.

Another area of human activity in which matrices have also found great application is the modeling of 3D images. Such tools are integrated into modern packages for implementing 3D models and allow designers to produce quickly and accurately the necessary calculations. The most striking representative of such systems is Compass-3D.

Another program, in which tools for such calculations are integrated, is Microsoft Office, and more specifically, an Excel spreadsheet.