More mathematics in ancient China used in their calculations entry in the form of tables with a certain number of rows and columns.Then, like mathematical objects referred to as a "magic square".Although the known uses of the tables in the form of triangles, which have not been widely adopted.
Today a mathematical matrix is understood obёkt rectangular shape with a predetermined number of columns and symbols that define the dimensions of the matrix.In mathematics, this notation has been widely used for recording systems in the compact form of the differential and linear algebraic equations.It is assumed that the number of rows in the matrix is equal to the number present in the system of equations correspond to the number of columns as necessary to determine the unknowns in the solution of the system.
addition, that in itself the matrix during its solution leads to finding the unknown, the condition laid down in the system of equations, there are a number of algebraic operations that are permitted to carry over a given mathematical object.This list includes the addition of matrices having the same dimensions.Multiplication of matrices with appropriate dimensions (it is possible to multiply a matrix with one side having a number of columns equal to the number of rows of the matrix on the other side).It is also permissible to multiply a matrix by a vector, or on a field element or the base ring (otherwise scalar).
Considering matrix multiplication, should be closely monitored, the number of columns to the first strictly corresponded to the number of rows of the second.Otherwise, the action of the matrix will be determined.According to the rule, by which the matrix-matrix multiplication, each element in the new array is equal to the sum of products of the corresponding elements of the rows of the first matrix elements taken from the other columns.
To illustrate, consider an example of how the matrix multiplication.Take the matrix A
2 3 -2
3 4 0
-1 2 -2,
multiply it by the matrix B
3 -2
0 1 4 -3.
the first row of the first column of the resulting matrix is equal to 2 * 3 + 3 * 1 + (- 2) * 4.Accordingly, in the first row in the second column is an element of 2 * (- 2) + 3 * 0 + (- 2) * (- 3), and so on until filling of each element of the new matrix.The rule of matrix multiplication requires that the result of the work of the matrix with the parameters in the mxn matrix having a ratio nxk, becomes a table which has a size of mx k.Following this rule, we can conclude that the work of the so-called square matrices, respectively, of the same order is always defined.
from the properties held by the matrix multiplication, should be distinguished as one of the basic fact that this operation is not commutative.That is the product of the matrix M to N is not equal to the product of N in M. If in square matrices of the same order is observed that their direct and inverse product is always identified, differing only in the result, the rectangular matrix similar condition of certainty is not always done.
matrix multiplication have a number of properties that have a clear mathematical proofs.Associativity multiplying means fidelity following mathematical expression: (MN) K = M (NK), where M, N, K, and - a matrix having the parameters at which the multiplication is defined.Distributivity multiplication suggests that M (N + K) = MN + MK, (M + N) K = MK + NK, L (MN) = (LM) N + M (LN), where L - number.
consequence of the properties of matrix multiplication, called "associative", it follows that in a work containing three or more factors, allowed entry without the use of brackets.
Using the distributive property makes it possible to disclose the brackets when considering matrix expressions.Please note, if we open the brackets, it is necessary to preserve the order of the factors.
Using matrix expressions not only compact record cumbersome systems of equations, but also facilitates the processing and decision.
