A matrix is a table that is populated with a specific set of numbers in a specific order. This term was coined by the prominent English scientist and theorist James Sylvester. He is one of the founders of the theory of application of these mathematical elements.
To date, they have found wide application in various calculations, which are based on such a method as, for example, finding the inverse matrix in various branches of human activity. This method is based on the determination of unknown parameters of a system of various equations and is often used in economic calculations.
There are the following special cases of these mathematical components: lowercase, columnar, zero, square, diagonal, unit. A row consists of only one row of elements, and a column consists of one column of numbers. Zero - all its elements are equal to 0. In a square such mathematical element, the number of columns is equal to the number of rows. In turn, in the diagonal, the elements located on the main diagonal are different from "0", and the rest in it should be equal to "0". Single is one of the subspecies of the diagonal matrix. She has only “1” on the main diagonal.
Matrix Examples:
where: A k is the general designation, a ij are the elements,
(a) –2nd order;
(b) - lowercase;
(c) –3rd order;
(d) is an example of a unit table of the 2nd order;
There is also an inverse matrix, the definition of which is as follows. When multiplied by the original table, the inverse is one. Many methods have been developed that provide the inverse matrix. The simplest of them is based on the definition of algebraic complements and a determinant (it is also sometimes called a determinant).
The determinant of a matrix is the expression a
11 a
22 -a
12 a
21, it is denoted as follows: ǀǀ. The above formula is valid for the table corresponding to the second order. There are formulas for determinants of matrices of a higher order. A prerequisite for the existence of a determinant - the table should be square. In practice, this element of this theory is most often used with a procedure such as finding the inverse matrix.
The second important component with which you can 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
the minor. Essentially, this is an additional determinant that can be obtained by mentally deleting the row and column in which this element is located. For example, for a table corresponding to the second order, which is given earlier in the text, the element a
11 will have the element a
22 as an algebraic complement.
Finding the inverse matrix is performed in 3 stages. At the first stage, the determinant is determined. The next step is all algebraic additions, which are then written in accordance with their indices, and we get a table of algebraic additions. At the final stage, an inverse matrix is obtained, the finding of which ends with the multiplication of each algebraic complement by a 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 convenient
form for
perception.Another area of human activity in which matrices have also found great use is the modeling of 3D images. Such tools are integrated into modern packages for the implementation of 3D-models and allow designers to quickly and accurately make the necessary calculations. The most striking representative of such systems is Compass-3D.
Another program that integrates tools for performing such calculations is Microsoft Office, and more specifically, the Excel spreadsheet processor.