Higher_Mathematics_Part_1

Micromodule 2

BASIC THEORETICAL INFORMATION. MATRICES

The definition of a matrix, operations with matrices. Inverse matrix. Matrix equations. Rank of a matrix and its properties.

Literature: [1, chapter 2—3], [4, part 2, p.p.2.2—2.4], [6, chapter 1, §2], [7, chapter 2, §7], [10, chapter 1, §2], [11, chapter 1, §1].

2.1. The basic concepts

Definition 1.6. A rectangular table of numbers written in the form

where aij are elements of this matrix.

Elements of a matrix form rows and columns. The first index i shows the number of the row and the second index j shows the number of the column

whose intersection is occupied by the element aij . For example, we have a matrix as

This matrix has three rows and three columns.

The product m × n is called the matrix dimension.

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For any square matrix

elements occupying the same places are equal, i.e. if aij = bij for all i and j (in

this case the number of rows (and, similarly, columns) of the matrices A and B must be equal).

2.2. Operations with matrices

The notation is A + B = C.

2. The product of the matrix A = (aij ) by any real number λ is a matrix

whose every element is equal to the product of the respective element of the matrix A by the number λ, i.e.

λA = λ(aij ) = (λaij ) (i = 1, 2, ..., m, j = 1, 2, ..., n).

3. The product of the matrix A = (aij ) , which has m rows and k columns by the matrix B = (bij ) which has k rows and n columns is a matrix С = (сij )

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which has m rows and n columns and whose element сij is equal to the sum of

the products of the elements of the i-th row of the matrix A by the j-th column of the matrix B, i.e.

In this the number k of columns of the matrix A must be equal to the number of rows of the matrix B, otherwise the product is not defined.

Remark. In general case AB ≠ BA . The multiplication of matrices is not commutative.

Properties of operations with matrices

2.3. Inverse matrix

An inverse matrix A−1 exists if and only if matrix A is non-singular А ( det(A) ≠ 0 ).

Definition 1.16. The inverse matrix A−1 is a matrix which satisfies the conditions

A−1 A = A A−1 = I,

were I is an identity matrix

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The inverse matrix can be calculated by the formula

where Aij is the cofactor of the element aij .

Remark. Pay the special attention to the order of indexes.

2.4. Matrices equations

Let’s consider the system of n linear equations with n unknowns

We write the above equations in the matrix form. We introduce the following designations

The matrix A is called the matrix of system. X is the matrix of unknowns. B is the matrix of constant terms.

Then applying the rule of multiplication of matrices the given system can be written by one matrix equation

A X = B.

Solution.

Let the matrix A is non-singular matrix (det A ≠ 0) . Therefore the inverse matrix A−1 exists. Multiply both parts of the matrix equation by A−1 we get

A−1 A X = A−1 B, (property A−1 A = I );

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I X = A−1 B, (property I X = X ).

We have

X = A−1 B.

Consequence. To decide a system it is sufficient to find the inverse matrix of system and multiply it by the matrix of constant terms B on the right.

2.5. Rank of a matrix

Let’s consider a matrix A of m × n

We choose in the given matrix A k rows and k columns, where number k ≤ m and k ≤ n. The determinant of order k, which consists of the cross elements

of chosen rows and columns, is called a minor of order k of the matrix A.

Definition 1.17. Rank of matrix A, designate as r(A) = rank(A), is called the greatest order of the minor if this minor isn’t equal to zero

1. The rank of a matrix exists for any matrix and

0≤ r(A) ≤ min(m, n).

2.r(A) = 0 if and only if A is zero matrix.

3.r(A) = n for square matrix A of order n if and only if the matrix A is non-

singular (det A ≠ 0).

For example, the matrix

We can calculate a rank of a matrix in the following way. If there is a minor of order k is not equal to zero and all minors of order (k+1) are equal to zero, then r(A) = k.

On the practice to find a rank of the highest orders it is useful to apply another method:

A rank of a matrix will not change if we use elementary transformations:

1)Interchanging of two rows (columns);

2)Multiplication of each element of a row (column) by a number k;

3)Addition elements of a row (column) and corresponding elements of another row (column) multiplying by a number k.

4)Elimination of a zero row from the matrix

Using elementary transformations a matrix can be transformed to the form

is equal to the number of rows of a matrix B. The operation of multiplication A B has the sense and we calculate the product of matrices in the following way:

then

f (A) = a0 An + a1 An−1 + an−1 A + an E .

Example 3. Find the inverse matrix, if

Solution. Firstly we calculate the determinant of matrix А:

Then the inverse matrix exists because А is a non-singular matrix. We find cofactors:

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So, the inverse matrix can be written in the form