The properties of the matrix and its determinant

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properties of matrices - a question that many may cause difficulty.Therefore it is necessary to consider it in detail.

Matrix - is a rectangular table of species, including the number and elements.Also, this kind of set of numbers and the elements of any other structure which is recorded as a rectangular table consisting of a certain number of rows and columns.This table must be enclosed in parentheses.This may be rounded brackets, such brackets or square brackets double direct type.All the numbers in the matrix are called - the matrix element and they have their coordinates in the table.Matrix compulsorily designated by a capital letter of the alphabet.

properties of matrices and mathematical tables include several aspects.Addition and subtraction of matrices passes strict element-wise.Multiplication and division goes beyond their normal arithmetic.To multiply one matrix to another, it is necessary to recall the information on the scalar product of one vector to another.

C = (a, b) = 1 and b 1 + a 2 2 b ... + and N b N

Properties of matrix multiplication are some nuances.The product of one matrix to another is non-commutative, that is, (a, b) does not equal (a, b).

The basic properties of matrices included such a thing as a measure of decency.A measure of decorum for such tables is considered to be the determinant.Determinant - it's kind of a function of several elements of a square matrix, a member of the order of n.In other words, the determinant is called determinants.A table with the second order determinant is equal to the difference between the product of the numbers or the elements of the two diagonals of the matrix-A11A22 A12A21.The determinant of the matrix with a higher order determinants expressed its blocks.

To understand how degenerate matrix was introduced such a thing as rank (rank) of the matrix.Rank - is the number of linearly independent columns and rows of the table.The matrix can be inverted only when it is full rank, ie rank (A) is equal to N.

Properties determinants of matrices include:

1. For the determinant of a square matrix will not change during its transposition.That is the determinant of this matrix is ​​the determinant of the amount to the table in transposed form.

2. If any column, or any string will include all zeros, then the determinant of such a matrix will be set to zero.

3. If any two columns of a matrix, or any two rows are interchanged, the sign of the determinant of such a table will change to the opposite.

4. If any column or any row of the matrix is ​​multiplied by any number, and its determinant is multiplied by this number.

5. If any element of the matrix is ​​written as the sum of two or more components, the determinant of this table is written as the sum of several determinants.Each determinant of such amount - is the determinant of a matrix, in which instead of the element represented by the amount recorded one of the terms of this amount, respectively priority determinant.

6. When a matrix has two rows with identical elements or two of the same column, the determinant of this table is equal to zero.

7. Also determinant is equal to zero at such a matrix, which has two columns and two rows are proportional to each other.

8. If the elements of a row or column multiplied by any number, and then add them to the elements in a different row or column of the same matrix, respectively, the determinant of the table will not change.

In total, we can say that the properties of the matrix is ​​a set of complex, but at the same time, the necessary knowledge about the nature of mathematical units.All properties of the matrix depends on its components and features.