Even the mathematicians of ancient China used in their calculations records in the form of tables with a certain number of rows and columns. Then, such mathematical objects were referred to as "magic squares." Although there are known cases of using tables in the form of triangles, which have not yet become widespread.
Today, by a mathematical matrix it is customary to understand an object of a rectangular shape with a given number of columns and characters, which determine the size of the matrix. In mathematics, this form of writing is widely used to write in a compact form systems of differential as well as linear algebraic equations. It is accepted that the number of rows in the matrix is ​​equal to the number of equations present in the system, the number of columns corresponds to how many unknowns must be determined during the solution of the system.
In addition to the fact that the matrix itself in the course of its solution leads to the finding of unknowns laid down in the condition of the system of equations, there are a number of algebraic operations that can be performed on this mathematical object. This list includes the addition of matrices having the same dimensions. Multiplication of matrices with suitable sizes (you can only multiply a matrix, on the one hand having the number of columns equal to the number of rows of the matrix on the other hand). It is also allowed to multiply the matrix by a vector, or by an element of the field or the main ring (otherwise a scalar).
When considering matrix multiplication, you should carefully monitor that the number of columns of the first strictly matches the number of rows of the second. Otherwise, this action on matrices will not be defined. According to the rule by which the matrix is ​​multiplied by the matrix, each element in the new matrix is ​​equated to the sum of the products of the corresponding elements from the rows of the first matrix by the elements taken from the columns of the other.
For clarity, consider an example of how matrix multiplication occurs. Take the matrix A
2 3 -2
3 4 0
-1 2 -2,
multiply it by matrix B
3 -2
10
4 -3.
The element of the first row of the first column of the resulting matrix is ​​2 * 3 + 3 * 1 + (- 2) * 4. Accordingly, in the first row in the second column there will be an element equal to 2 * (- 2) + 3 * 0 + (- 2) * (- 3), and so on until each element of the new matrix is ​​filled. The matrix multiplication rule assumes that the result of the product of the matrix with the parameters mxn and the matrix having the ratio nxk will be a table that has dimensions m x k. Following this rule, we can conclude that the product of the so-called square matrices of the same order, respectively, is always defined.
Of the properties possessed by matrix multiplication, it should be singled out as one of the main that this operation is not commutative. That is, the product of the matrix M by N is not equal to the product of N by M. If it is observed in square matrices of the same order that their direct and inverse products are always determined, differing only in the result, then for rectangular matrices such a certainty condition is not always fulfilled.
Matrix multiplication has a number of properties that have clear mathematical proofs. The associativity of multiplication implies the correctness of the following mathematical expression: (MN) K = M (NK), where M, N, and K are matrices having parameters for which multiplication is defined. The distributive nature of the multiplication assumes that M (N + K) = MN + MK, (M + N) K = MK + NK, L (MN) = (LM) N + M (LN), where L is a number.
As a consequence of the property of matrix multiplication called “associativity”, it follows that in a work containing from three or more factors, recording without the use of brackets is allowed.
Using the distributive property makes it possible to open brackets when considering matrix expressions. Please note that if we open the brackets, then we need to preserve the order of the factors.
Using matrix expressions allows not only compact recording of bulky systems of equations, but also facilitates the process of processing and solving them.