Matrix calculator online

A rectangular matrix of size mxn is the set of mn numbers arranged in a rectangular table containing m rows and n columns. We write the matrix in the form

A = (4.1)

or abbreviated as A = (aij) (i =; j =). The numbers aij, constituting the matrix are called its elements, the first index indicates the row number, second - on the column. Two matrices A = (aij) and B = (bij) are the same size are called equal if their elements are pairwise equal standing in the same field, ie A = B, if aij = bij.

Matrix, consisting of one row or one column, called respectively a row vector or column vector. Column vectors and row vectors is simply called vectors.

Matrix consisting of a single number, is identified with that number. Matrix of size mxn, all of whose elements are equal to zero, called the zero matrix and is denoted by 0. Elements of the matrix with the same indices are called elements of the main diagonal. If the number of rows of the matrix equals the number of columns, ie m = n, is called a square matrix of order n. Square matrix whose only nonzero elements of the main diagonal, are called diagonal matrices and written as follows:

.

If all the elements aii diagonal matrix calculator online equal to 1, then the matrix is called the unit and is denoted by the letter E:

E =.

A square matrix is called triangular if all elements of standing up (or down) the main diagonal are zero. Transposition is a transformation matrix online calculator, where rows and columns are swapped with the preservation of their rooms. Denotes transposition symbol T at the top.

Given a matrix (4.1). Interchange rows with columns. Obtain matrix

AT =,

be transposed to the matrix A. In particular, when transposing the column vector a vector-line and vice versa.

Product of the matrix A by the number λ is called a matrix whose elements are obtained from the relevant elements of the matrix A by multiplying by the number λ: λA = (λaij).

The sum of two matrices A = (aij) and B = (bij) of the same size is the matrix C = (cij) of the same size whose elements are determined by the formula cij = aij + bij.

The product AB of A by the matrix B is determined by the assumption that the number of columns of the matrix A equals the number of rows of B.

The product of two matrices A = (aij) and B = (bjk), where i =, j =, k =, defined in a specific order AB is a matrix C = (cik), whose elements are defined by the following rule:

cik = ai1b1k + ai2b2k + ... + Aimbmk = aisbsk. (4.2)

In other words, elements of the matrix-product is defined as follows: the element of i-th row and k-th column of the matrix C is equal to the sum of products of elements of the i-th row of the matrix A to the relevant elements of the k-th column of the matrix B.
2. Determinants

Permutation of numbers 1, 2 ,..., n is any arrangement of these numbers in a certain order. In elementary algebra, we prove that the number of permutations that can be formed from n numbers is 12 ... n = n!. For example, the three numbers 1, 2, 3 can form 3! = 6 permutations: 123, 132, 312, 321, 231, 213. They say that in this permutation of i and j form an inversion (disorder), if i> j, but i should in this rearrangement before j, ie if more is left less.

A permutation is called even (or odd) if it accordingly even (odd) total number of inversions. The operation, through which pass from one permutation to another, composed of the same n numbers is called the substitution of n-th degree.

Substitution that transforms one permutation into another, written two rows in general brackets, with numbers, occupying the same space in these rearrangements are called relevant and are written one above the other. For example, the symbol denotes the substitution, in which 3 goes into 4, 1 → 2, 2 → 1, 4 → 3. Substitution is called even (or odd) if the total number of inversions in the two rows of substitution is even (odd). Any permutation of the n-th degree can be written in the form, ie with natural arrangement of numbers in the top row.

Suppose we are given a square matrix of order n

. (4.3)

Consider all the possible products of n elements of this matrix, taken one by one and only one from each row and each column, ie products of the form:

, (4.4)

where the indices q1, q2 ,..., qn are some permutation of the numbers

1, 2 ,..., n. The number of such works is the number of different permutations of n symbols, ie equal to n!. Sign of the product (4.4) is (- 1) q, where q - the number of inversions in the permutation of the second index elements.

The determinant of n-th order, the corresponding matrix (4.3), is called the algebraic sum of n! terms of the form (4.4). To write the determinant of the symbol is used, or A = det A = (determinant or determinant of the matrix A).

Properties of determinants

1. Determinant is not changed by transposition.

2. If one of the rows of the determinant consists of zeros, the determinant is zero.

3. If you rearrange the determinant of the two lines, the determinant will change the sign.

4. Key, containing two identical rows, equal to zero.

5. If all the elements of a row of the determinant is multiplied by some number k, then the determinant itself is multiplied by k.

6. Key to containing two proportional rows is zero.

7. If all elements of the i-th row of the determinant represented as the sum of two terms aij = bj + cj (j =), the determinant is the sum of determinants in which all lines except the i-th - the same as in a given determinant, and i-th row in one of the components consists of the elements bj, in the other - of the elements cj.

8. Determinant does not change if the elements of one of his lines are added the relevant elements of another row, multiplied by one and the same number.

Note. All the properties remain valid if instead of lines to take the columns.

Minor Mij element aij determinant d n-th order is the determinant of order n-1, which is obtained from d by deleting the row and column containing the item.

Cofactor of the element aij determinant d is called a minor Mij, taken with the (-1) i + j. Cofactor of the element aij denote Aij. Thus, Aij = (-1) i + j + Mij.

Ways of practical computing determinants, based on the fact that the determinant of order n can be expressed through the determinants of lower orders, gives the following theorem.
Theorem (expansion of the determinant along a row or column).

The determinant is the sum of all elements of arbitrary products of its row (or column) by their cofactors. In other words, we have the decomposition d of elements of i-th row

d = ai1Ai1 + ai2Ai2 + ... + AinAin (i =)

or j-th column

d = a1jA1j + a2jA2j + ... + AnjAnj (j =).

In particular, if all elements of the row (or column), except one, are equal to zero, the determinant is equal to this element, multiplied by its cofactor.
3. Rank matrix

Consider a rectangular matrix (4.1). If in this matrix to allocate an arbitrary k rows and k columns, elements at the intersection of selected rows and columns form a square matrix of k-th order. The determinant of this matrix is called a minor of the k-th order of the matrix A. It is obvious that the matrix A has the minors of any order of 1 to the smallest of the numbers m and n. Among all non-zero minors of the matrix A has at least one minor, the order of which will be the greatest. Most of the orders of the minors of this matrix, different from zero, is called the rank of the matrix. If the rank of A is r, then this means that the matrix A has a nonzero minor of order r, but every minor of order greater than r, is zero. Rank of matrix A is denoted by r (A). Is obvious that the relation

0 ≤ r (A) ≤ min (m, n).

The rank of the matrix is either by the halo of the minors, either by elementary transformations. In calculating the rank of the first way to make the transition from the minors to the lower order minors of higher order. If you have already found minor D k-th order matrix A is different from zero, the calculations require only minors (k +1)-th order, bordering minor D, ie containing it as a minor. If they are equal to zero, then the rank is equal to k.

Elementary transformation matrices are called as follows:

1) permutation of any two rows (or columns)

2) multiplication of rows (or columns) on a non-zero number,

3) additions to a single row (or column) of another row (or column) multiplied by some number.

Two matrices are called equivalent if one of them is obtained from the other by 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, it is written as: A ~ B.

Canonical matrix is a matrix, whose early

main diagonal row a few units (the number

may be zero), and all other elements equal to zero,

eg.

With the help of elementary transformations of rows and columns of any matrix can be reduced to canonical. The rank of a canonical matrix equals the number of units on its main diagonal.
4. Inverse matrix

Consider a square matrix

A =.

Let Δ = det A.

A square matrix A is called nondegenerate or nonsingular if its determinant is nonzero, and degenerate, or especially, if Δ = 0.

A square matrix B is called the inverse of the square matrix A of the same order, if their product A B = A = E, where E - the identity matrix of the same order as the matrix A and B.

Theorem. To make the matrix A had the opposite, it is necessary and sufficient that its determinant be nonzero.

Inverse of matrix A, denoted by A-1, so that B = A-1. The inverse matrix is calculated by the formula

A-1 = 1 / Δ, (4.5)

where Aij - cofactors of the elements aij.

Computation of the inverse matrix formula (4.5) for matrices of high order is very time consuming, so in practice it is convenient to find the inverse matrix using the method of elementary transformations (VC). Any nonsingular matrix A by VC only columns (or rows only) can lead to the identity matrix E. If committed over the matrix A VC in the same manner applied to the unit matrix E, then the result is the inverse matrix. It is convenient to perform EP on matrices A and E at the same time, recording both the matrix number in line. We note again that in determining the canonical form of the matrix to find its rank can use the transformations of rows and columns. If you want to find the inverse matrix, in the process of transformation should be used only rows or only columns.