Math Higher Mathematics   Solving tasks and exercises - OnLine
Language : Russian

# Matrices

Matrixis like a table. Matrix - a set of numbers or any other objects arranged in a form of rectangular table:
A =
 a11 a12 ... a1n a21 a22 ... a2n ... ... ... ... am1 am2 ... amn
This table consists of m line and n columns and contains mn cells (positions). Then it is said that the matrix has size m x n and it is called the (m x n)-matrix. Position at the cross of i-th row and j-th column is called ij-cell.

Numbers or any other objects at the table cells are called elements of the matrix. Location of elements is strictly fixed: in each cell must be placed only one elementand no cell should not remain free. In general designation, the element's aij first index i always indicates the number of lines, and the second - the number of the column. Element located at the ijcell, is called the ij-element.

Matrix is signed with a single letter (often letters for matrices are bold type or can supply any additional characters). However, regardless of this the matrix is always a set of table-ordered elements. Two matrices are equal only if their corresponding elements are equal, e.g., А = В as long as aij = bij (i = 1,2, ... , m; j = 1,2, ... , n). It is clear that one can only compare the matrices of the same size with the elements of the same nature

Matrix whose elements are real or complex numbers, called respectively real or complex. А is a complex (m x n)-matrix with akj = rkj+ibkj. The matrix А* of the same size with the elements a*kj = rkj-ibkj is called the complex conjugate of А.

Often, the zero elements in the table are not written down, but keep in mind that the empty cells also contain numbers (zeros).

In addition to the above "cell representation", there're other ways to represent matrices, e.g.: Matrices first appeared in the middle of XIX century in the works of english mathematicians A.Keli andU.Gamiltona. Representation of a complex of elements in a matrix turned out very useful and it is widely used in physics, engineering, and economics.

To the list of possible tasks...