Is the minor of .
Definition
3. The
algebraic complement of an element аij
is the minor of aij
multiplied by -1 raised to the proper equal to the sum of the numbers
of the row and the column intersecting in the given element:
.
9. A determinant equals the sum of products of all elements of any lines and the corresponding algebraic complements.
.
For a kth order determinant, we can write property 9 in the form of expansion along the kth-column:
.
10. The sum of the products of the elements of any line and the algebraic complements of the corresponding elements of a parallel line equals zero:
=
,
if m
k.
Combining properties 9 and 10, we obtain
where
is the Kronecker symbol.
§2. Matrices and operations on them. Inverse matrix
Let m and n be natural numbers, A rectangular
array of numbers
enclosed by brackets is called a matrix:
(2)
The numbers that make up a matrix are called the
elements or entries. The number of rows (horizontal) and columns
(vertical) determine the dimensions of the matrix. Or abbreviated
where i –
row number; j
- column number.
Definition 4. (Definition of a matrix)
Rectangular array of real numbers
m rows by n columns
Named using capital letters
First subscript is row, second subscript is column
Terminology
A matrix with m rows and n columns is called a matrix of order m x n.
A square matrix is a matrix with an equal number of rows and columns. Since the number of rows and columns are the same, it is said to have order n.
The main diagonal of a square matrix are the elements from the upper left to the lower right of the matrix.
A row matrix is a matrix that has only one row.
A column matrix is a matrix that has only one column.
A matrix with only one row or one column is called a vector.
Principal diagonal - the diagonal in a square matrix that runs from upper left to lower right.
Two matrices A and B of the same size – the same number of rows and the same number of columns – are called equal provided that each element of A is equal to the corresponding element of B. Thus two matrices of the same size are equal provided they are elementwise equal, and we write A=B to denote equality of the two matrices A and B.
Types of Matrices
An n x n matrix is called a square matrix of order n.
An n x m matrix is called a rectangular matrix of order n x m.
A matrix whose all entries are zero is called zero matrix (null-matrix).
An n x m matrix whose main diagonal (from upper left to lower right) has all elements 1 while all the other elements are 0 is called an identity matrix , and is denoted by I n x m .
A square matrix is called a diagonal matrix if all entries that are not on the main diagonal are zero.
In a square matrix , either upper left or lower right part of the main diagonal has all elements are zero is called a triangular matrix.
6) a zero matrix or null matrix is a matrix with all its entries being zero. Some examples of zero matrices are
Definition 5. Matrices of the same size are equal if their respective elements are equal. Two matrices of the same size can be added (elementwise):
.
Any matrix can be multiplied by any number all elements of the matrix should be multiplied by this number):
.
Subtraction is defined as
А–В=А+(–1)В.
Two matrices can
be multiplied only if the number of
columns
in
the first matrix equals the number
of rows in
the second matrix. An
matrix is multiplied by an
matrix
as follows:
We obtain
a
matrix of size
.
Multiplication of matrices is not commutative, and sometimes even impossible:
АВВА.
Definition 6. A square matrix is said to be non-degenerate, if it has nonzero determinant.
Consider the identity matrix
.
Definition 7. The inverse matrix for А is the matrix А–1for which
или
.
Inverse matrices exist only for non-degenerate matrices.
The method for finding the inverse matrix.
Consider a square non-degenerate matrix of size n
n
.
Let us compose the square matrix
,
where the Аij are the algebraic complements of аij. The matrix А* is called the adjoint matrix.
Let us multiply А by А* :
The remaining elements of the first row in the new matrix are the sum of the products of the elements of the first row and the algebraic complements of the parallel rows. According to property 10, they equal zero. The same is true for the elements of the second row, except the second element, which are the sum of the products of the elements of the second row by the algebraic complements of the first, third , ... rows.
To obtain the identity matrix from this diagonal
matrix, we must divide it by the determinant
,
i.e.,
;
consequently,
,
,
.