Linear algebra

1. Linear Algebra

1.1 Matrices. Basic concepts. Matrix Operations

A matrix of size mn is the set of any elements arranged in the form of a rectangular table consisting of m-rows of n-columns provided by algebra homework help service.

The matrix is ​​denoted by capital letters of the Latin alphabet A, B, C, ....

Here aij are the elements of the matrix. Each element has two indexes, the first i denotes the row number (i = 1, 2,, m), and the second j denotes the column number. (j = 1, 2,, n). Elements of the matrix can be numbers, functions, vectors, etc.

Consider some types of matrices.

If m = n, then the square matrix is ​​of order n

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A single row matrix is ​​called a row matrix

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A single column matrix is ​​called a column matrix.

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A square matrix in which all elements not standing on the main diagonal are equal to zero is called the diagonal

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The diagonal matrix in which all elements are equal to unity is called the identity matrix

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If in the matrix A to swap rows and columns, then the resulting matrix is ​​called transposed.

then.

Two matrices A and B are equal to each other if they are of the same size and their corresponding elements are equal, i.e.

A = B, if aij = bij (i = 1,2, ... m; j = 1,2, ..., n).

Matrix Addition

Only matrices of the same size can be added according to the rule

,,,

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Example. Find the matrix if

,.

Decision.

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Matrix Addition Properties

A + B = B + A;

A + (B + C) = (A + B) + C = A + B + C.

Multiplication of a matrix by a number

To multiply the matrix by a number, you need to multiply by this number each element of the matrix.

, B = A,.

Properties of matrix multiplication by a number

,,

(+) A = A + A,

() A = (B).

Example. Find the matrix if

Decision.

Example. Find the matrix if

.

Decision.

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The product of two matrices

You can only multiply those matrices for which the number of columns in the first matrix is ​​equal to the number of rows in the second matrix. The product of two matrices

,

called matrix

,

whose element cij is found by the formula

, i = 1,2, ..., m; j = 1,2, ..., p,

those. the element of the matrix cij at the intersection of the i-row and j-column is equal to the sum of the products of the i-row elements of the matrix A by the corresponding elements of the j-column of the matrix B. As a result of multiplying the matrix A by the matrix B, the matrix C is obtained, the number of rows of which is equal to the number rows of matrix A, and the number of columns is equal to the number of columns of matrix B.

Example. Multiply matrices A and B if

,.

Decision.

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If AB = VA, then the matrices are commutative.

1.2 Determinants and their properties

The determinant of a square matrix or simply the determinant (determinant) is a number that is put into correspondence with the matrix and can be calculated by its elements.

A square matrix of the first order consists of one element, so its determinant is equal to the element itself.

The second-order determinant is calculated by the formula:

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The third-order determinant is calculated according to the triangle rule:

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The minor Mij of the element aij of the determinant of the nth order is the determinant of the (n 1) th order obtained from this determinant by deleting the elements of the ith row and jth column.

The algebraic complement Aij of the element aij of the determinant is the number Aij = (1) i + jMij.

Thus, the algebraic complement Aij of aij is the corresponding minor Mij multiplied by.

Calculation of determinants

Theorem (without proof) on the decomposition of the determinant into elements of a row (column). For each square matrix A of order n, the formula

, if ;

, if .

Example. ,

Qualifier Properties

1. When transposing, the value of the determinant does not change.

The rows and columns of the qualifier are equivalent.

2. If in determinants to swap any two rows (columns) in places, then the determinant changes sign.

3. The determinant with two identical columns (rows) is equal to zero.

4. When multiplying the elements of a column (row) by a number, the determinant is multiplied by this number.

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5. If all elements of a column (row) are equal to zero, then the determinant is equal to zero.

6. If the elements of two rows (columns) are proportional, then the determinant is zero.

7. Let each element of any column (row) of the determinant be equal to the sum of two terms, then this determinant is equal to the sum of two determinants, and in the first of them the corresponding column (row) consists of the first terms, and in the second of the second terms.

8. The determinant will not change if the corresponding elements of another column multiplied by the same number are added to the elements of a column (row).

Where

9. The sum of the products of the elements of any determinant column by the algebraic complement to the elements of another column is zero.

Example. Calculate the determinant

2. The expansion on the first line:

3. Convert the first column:

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