Theme 9. Matrix operations in Maple. Systems of linear equations

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 number a _{ij is} called its elements, the first index denotes the line number, the second is the number of the column, at the intersection of which there is an element (i = 1, 2, ..., m; j = 1,
2, ..., n.

The matrix is ​​called the expression, where
 - the number of rows, and - the number of columns of the matrix.

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

A single matrix is ​​a square matrix in which the diagonal elements are equal, and all the last ones.

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 that is obtained from the matrix, 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 product of a matrix on an arbitrary number is called a matrix whose elements are the product of the matrix elements on:

The sum (difference) of two identical dimensions n matrices is called the matrix, the elements of which are equal to the sum (s) of the corresponding matrix elements, and, that is, for the sum of the matrices and for the difference of the matrices, where the elements of the matrix, the elements of the matrix

The product of the dimension matrix dimension matrix is ​​called the matrix, the elements of which are equal to the sum of the products of the elements of the ith row of the matrix on the corresponding elements of the jth column of the matrix.

The product of the matrices is noncommutative:.

For any matrix there is a single matrix such that the sum of these matrices is equal to the zero matrix:.

The opposite matrix is ​​denoted by.

The difference between two matrices of the same size is a matrix for which equality is true. The difference always exists and equals.

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