Lesson 9

Matrix (m × n) is a rectangular two-dimensional table that includes rows and columns of elements that can be a number, a variable, a symbol, an expression.

In the matrix, the numbers a _ij are called its elements, the first index denotes the line number, the second is the column number at the intersection of the element (i = 1, 2, ..., m; j = 1, 2, ..., n) .

The dimension of the matrix is ​​called the expression n × m , where n is the number of rows, and m is the number of columns of the matrix.

Square matrix - The number of rows is equal to the number of columns.

The single matrix E is a square matrix in which the diagonal elements are equal to 1, and all the last 0.

Transposed matrix - columns and rows change cities.

An inverse matrix - a matrix, multiplied by an output matrix, gives a unit matrix.

A matrix is ​​called zero if all its elements are zero.

The matrix At, which is obtained from matrix A, if all the rows of the matrix are replaced by the corresponding columns, is called transposed .

The square matrix, in which all the elements outside the main diagonal are zero, is called diagonal.

Actions on matrices

The matrix product for an arbitrary number is a matrix.

The sum (difference) of two identical dimensions n matrices A and B  is a matrix whose elements are equal to the sum (s) of the corresponding elements of the matrices A and B , that is, with _andј = a _andј + b _ij for the sum of the matrices and with _іј = а _іј - b _ij difference matrices for ( i = 1, 2, ..., m ; j = 1, 2, ..., n ) , where a _iј  - elements of the matrix A , b _ij - elements of matrix B.

The product of the matrix of the dimension k × s of the matrix A of the dimension s × n is k × n - the matrix, the elements of the SIU (i = 1, 2, ... k; j = 1, 2, ..., n) of which are the sum of the items of the elements ith row of matrix B to the corresponding elements of the jth column of matrix A.

The product of the matrices is noncommutative  A • B ≠ B • A.

Opposite matrix denoting - A = (- 1) • A .

The difference between two matrices A of the same size is a matrix C , for which the equality B + C = A is true . The difference always exists and equals A + (- B) .

The identifier of a square matrix is ​​called is a number.