Finding the rank of a matrix by the method of elementary transformations. Calculating the rank of a matrix using elementary transformations

Definition. Matrix rank is the maximum number of linearly independent rows considered as vectors.

Theorem 1 on the rank of a matrix. Matrix rank is the maximum order of a non-zero minor of a matrix.

We have already discussed the concept of a minor in the lesson on determinants, and now we will generalize it. Let's take some rows and some columns in the matrix, and this "something" should be less than the number of rows and columns of the matrix, and for rows and columns this "something" should be the same number. Then at the intersection of how many rows and how many columns there will be a matrix of a smaller order than our original matrix. The determinant of this matrix will be a k-th order minor if the mentioned "something" (the number of rows and columns) is denoted by k.

Definition. Minor ( r+1)-th order, inside which lies the chosen minor r-th order, is called bordering for the given minor.

The two most commonly used methods finding the rank of a matrix. it way of fringing minors and method of elementary transformations(by the Gauss method).

The method of bordering minors uses the following theorem.

Theorem 2 on the rank of a matrix. If it is possible to compose a minor from the elements of the matrix r th order, which is not equal to zero, then the rank of the matrix is equal to r.

With the method of elementary transformations, the following property is used:

If a trapezoidal matrix equivalent to the original one is obtained by elementary transformations, then the rank of this matrix is the number of lines in it except for lines consisting entirely of zeros.

Finding the rank of a matrix by the method of bordering minors

A bordering minor is a minor of a higher order in relation to the given one, if this minor of a higher order contains the given minor.

For example, given the matrix

Let's take a minor

edging will be such minors:

Algorithm for finding the rank of a matrix next.

1. We find minors of the second order that are not equal to zero. If all second-order minors are equal to zero, then the rank of the matrix will be equal to one ( r =1 ).

2. If there exists at least one second-order minor that is not equal to zero, then we compose bordering third-order minors. If all third-order bordering minors are zero, then the rank of the matrix is two ( r =2 ).

3. If at least one of the bordering minors of the third order is not equal to zero, then we compose the minors bordering it. If all bordering fourth-order minors are zero, then the rank of the matrix is three ( r =2 ).

4. Continue as long as the size of the matrix allows.

Example 1 Find the rank of a matrix

Solution. Minor of the second order .

We frame it. There will be four bordering minors:

Thus, all bordering third-order minors are equal to zero, therefore, the rank of this matrix is two ( r =2 ).

Example 2 Find the rank of a matrix

Solution. The rank of this matrix is 1, since all second-order minors of this matrix are equal to zero (in this, as in the cases of bordering minors in the next two examples, dear students are invited to verify for themselves, perhaps using the rules for calculating determinants), and among first-order minors , that is, among the elements of the matrix, there are not equal to zero.

Example 3 Find the rank of a matrix

Solution. The second-order minor of this matrix is , and all third-order minors of this matrix are zero. Therefore, the rank of this matrix is two.

Example 4 Find the rank of a matrix

Solution. The rank of this matrix is 3 because the only third order minor of this matrix is 3.

Finding the rank of a matrix by the method of elementary transformations (by the Gauss method)

Already in Example 1, it can be seen that the problem of determining the rank of a matrix by the method of bordering minors requires the calculation of a large number of determinants. There is, however, a way to reduce the amount of computation to a minimum. This method is based on the use of elementary matrix transformations and is also called the Gauss method.

Elementary transformations of a matrix mean the following operations:

1) multiplication of any row or any column of the matrix by a number other than zero;

2) adding to the elements of any row or any column of the matrix the corresponding elements of another row or column, multiplied by the same number;

3) swapping two rows or columns of a matrix;

4) removal of "null" rows, that is, those, all elements of which are equal to zero;

5) deletion of all proportional lines, except for one.

Theorem. The elementary transformation does not change the rank of the matrix. In other words, if we use elementary transformations from the matrix A go to matrix B, then .

Let some matrix be given:

Select in this matrix arbitrary lines and arbitrary columns
. Then the determinant th order, composed of matrix elements
located at the intersection of selected rows and columns is called a minor -th order matrix
.

Definition 1.13. Matrix rank
is the largest order of the non-zero minor of this matrix.

To calculate the rank of a matrix, one should consider all its minors of the smallest order and, if at least one of them is nonzero, proceed to the consideration of minors of the highest order. This approach to determining the rank of a matrix is called the bordering method (or the bordering minors method).

Task 1.4. By the method of bordering minors, determine the rank of a matrix
.

Consider first-order bordering, for example,
. Then we turn to the consideration of some bordering of the second order.

For example,
.

Finally, let's analyze the bordering of the third order.

So the highest order of a non-zero minor is 2, hence
.

When solving Problem 1.4, one can notice that the series of bordering minors of the second order are nonzero. In this regard, the following notion takes place.

Definition 1.14. The basis minor of a matrix is any non-zero minor whose order is equal to the rank of the matrix.

Theorem 1.2.(Basic minor theorem). Basic rows (basic columns) are linearly independent.

Note that the rows (columns) of a matrix are linearly dependent if and only if at least one of them can be represented as a linear combination of the others.

Theorem 1.3. The number of linearly independent matrix rows is equal to the number of linearly independent matrix columns and is equal to the rank of the matrix.

Theorem 1.4.(Necessary and sufficient condition for the determinant to be equal to zero). In order for the determinant -th order is equal to zero, it is necessary and sufficient that its rows (columns) be linearly dependent.

Calculating the rank of a matrix based on its definition is too cumbersome. This becomes especially important for high-order matrices. In this regard, in practice, the rank of a matrix is calculated based on the application of Theorems 10.2 - 10.4, as well as the use of the concepts of matrix equivalence and elementary transformations.

Definition 1.15. Two matrices
and are called equivalent if their ranks are equal, i.e.
.

If matrices
and are equivalent, then note
 .

Theorem 1.5. The rank of a matrix does not change from elementary transformations.

We will call elementary transformations of the matrix
any of the following actions on the matrix:

Replacing rows with columns and columns with corresponding rows;

Permutation of matrix rows;

Crossing out a line, all elements of which are equal to zero;

Multiplying any string by a non-zero number;

Adding to the elements of one row the corresponding elements of another row multiplied by the same number
.

Corollary of Theorem 1.5. If the matrix
obtained from the matrix using a finite number of elementary transformations, then the matrices
and are equivalent.

When calculating the rank of a matrix, it should be reduced to a trapezoidal form using a finite number of elementary transformations.

Definition 1.16. We will call trapezoidal such a form of representation of a matrix when in the bordering minor of the largest order other than zero, all elements below the diagonal ones vanish. For example:

Here
, matrix elements
turn to zero. Then the form of representation of such a matrix will be trapezoidal.

As a rule, matrices are reduced to a trapezoidal shape using the Gaussian algorithm. The idea of the Gaussian algorithm is that, by multiplying the elements of the first row of the matrix by the corresponding factors, they achieve that all elements of the first column located below the element
, would turn to zero. Then, multiplying the elements of the second column by the corresponding multipliers, we achieve that all elements of the second column located below the element
, would turn to zero. Further proceed similarly.

Task 1.5. Determine the rank of a matrix by reducing it to a trapezoidal form.

For the convenience of applying the Gaussian algorithm, you can swap the first and third rows.








.

Obviously here
. However, to bring the result to a more elegant form, further transformations over the columns can be continued.










.

>>Matrix rank

Matrix rank

Determining the rank of a matrix

Consider a rectangular matrix. If in this matrix we select arbitrarily k lines and k columns, then the elements at the intersection of the selected rows and columns form a square matrix of the kth order. The determinant of this matrix is called k-th order minor matrix A. Obviously, the matrix A has minors of any order from 1 to the smallest of the numbers m and n. Among all non-zero minors of the matrix A, there is at least one minor whose order is the largest. The largest of the non-zero orders of the minors of a given matrix is called rank matrices. If the rank of matrix A is r, then this means that the matrix A has a non-zero minor of order r, but every minor of order greater than r, equals zero. The rank of a matrix A is denoted by r(A). It is obvious that the relation

Calculating the rank of a matrix using minors

The rank of a matrix is found either by the bordering of minors, or by the method of elementary transformations. When calculating the rank of a matrix in the first way, one should pass from minors of lower orders to minors of a higher order. If a non-zero minor D of the kth order of the matrix A has already been found, then only the (k + 1)th order minors bordering the minor D must be calculated, i.e. containing it as a minor. If they are all zero, then the rank of the matrix is k.

Example 1Find the rank of a matrix by the method of bordering minors

Solution.We start with minors of the 1st order, i.e. from the elements of the matrix A. Let us choose, for example, the minor (element) М 1 = 1 located in the first row and the first column. Bordering with the help of the second row and the third column, we obtain the minor M 2 = , which is different from zero. We now turn to minors of the 3rd order, bordering M 2 . There are only two of them (you can add a second column or a fourth). We calculate them: = 0. Thus, all bordering minors of the third order turned out to be equal to zero. The rank of matrix A is two.

Calculating the rank of a matrix using elementary transformations

ElementaryThe following matrix transformations are called:

1) permutation of any two rows (or columns),

2) multiplying a row (or column) by a non-zero number,

3) adding to one row (or column) another row (or column) multiplied by some number.

The two matrices are called equivalent, if one of them is obtained from the other with the help of a finite set of elementary transformations.

Equivalent matrices are not, generally speaking, equal, but their ranks are equal. If the matrices A and B are equivalent, then this is written as follows: A~b.

CanonicalA matrix is a matrix that has several 1s in a row at the beginning of the main diagonal (the number of which may be zero), and all other elements are equal to zero, for example,

With the help of elementary transformations of rows and columns, any matrix can be reduced to a canonical one. The rank of a canonical matrix is equal to the number of ones on its main diagonal.

Example 2Find the rank of a matrix

and bring it to canonical form.

Solution. Subtract the first row from the second row and rearrange these rows:

Now, from the second and third rows, subtract the first, multiplied by 2 and 5, respectively:

;

subtract the first from the third row; we get the matrix

B = ,

which is equivalent to the matrix A, since it is obtained from it using a finite set of elementary transformations. Obviously, the rank of matrix B is 2, and hence r(A)=2. The matrix B can easily be reduced to the canonical one. Subtracting the first column, multiplied by suitable numbers, from all subsequent ones, we turn to zero all the elements of the first row, except for the first, and the elements of the remaining rows do not change. Then, subtracting the second column, multiplied by the appropriate numbers, from all subsequent ones, we turn to zero all the elements of the second row, except for the second, and get the canonical matrix:

Elementary The following matrix transformations are called:

1) permutation of any two rows (or columns),

2) multiplying a row (or column) by a non-zero number,

3) adding to one row (or column) another row (or column) multiplied by some number.

The two matrices are called equivalent, if one of them is obtained from the other with the help of a finite set of elementary transformations.

Equivalent matrices are not, generally speaking, equal, but their ranks are equal. If the matrices A and B are equivalent, then this is written as: A ~ B.

Canonical A matrix is a matrix that has several 1s in a row at the beginning of the main diagonal (the number of which may be zero), and all other elements are equal to zero, for example,

With the help of elementary transformations of rows and columns, any matrix can be reduced to a canonical one. The rank of a canonical matrix is equal to the number of ones on its main diagonal.

Example 2 Find the rank of a matrix

and bring it to canonical form.

Solution. Subtract the first row from the second row and rearrange these rows:

Now, from the second and third rows, subtract the first, multiplied by 2 and 5, respectively:

;

subtract the first from the third row; we get the matrix

B = ,

Kronecker - Capelli theorem- criterion of compatibility of the system of linear algebraic equations:

For a linear system to be compatible, it is necessary and sufficient that the rank of the extended matrix of this system be equal to the rank of its main matrix.

Proof (system compatibility conditions)

Need

Let system joint. Then there are numbers such that . Therefore, the column is a linear combination of the columns of the matrix. From the fact that the rank of a matrix will not change if the system of its rows (columns) is deleted or a row (column) is assigned, which is a linear combination of other rows (columns), it follows that .

Adequacy

Let . Let's take some basic minor in the matrix. Since , then it will also be the basis minor of the matrix . Then, according to the basis theorem minor, the last column of the matrix will be a linear combination of the base columns, that is, the columns of the matrix . Therefore, the column of free members of the system is a linear combination of the columns of the matrix.

Consequences

Number of main variables systems equal to the rank of the system.

joint system will be defined (its solution is unique) if the rank of the system is equal to the number of all its variables.

Homogeneous system of equations

Sentence15 . 2 Homogeneous system of equations

is always collaborative.

Proof. For this system, the set of numbers , , , is a solution.

In this section, we will use the matrix notation of the system: .

Sentence15 . 3 The sum of solutions to a homogeneous system of linear equations is a solution to this system. A solution multiplied by a number is also a solution.

Proof. Let and serve as solutions of the system . Then and . Let . Then

Since , then is a solution.

Let be an arbitrary number, . Then

Since , then is a solution.

Consequence15 . 1 If a homogeneous system of linear equations has a non-zero solution, then it has infinitely many different solutions.

Indeed, multiplying a non-zero solution by different numbers, we will get different solutions.

Definition15 . 5 We will say that the solutions systems form fundamental decision system if the columns form a linearly independent system and any solution to the system is a linear combination of these columns.