Properties of the matrix and its determinant

Properties of matrices - a question that many can cause difficulties. Therefore, it is worth considering it in more detail.

The matrix is a table of rectangular form, including numbers and elements. It is also a collection of numbers and elements of some other structure that are written as a rectangular table consisting of a certain number of rows and columns. Such a table must be enclosed in parentheses. These can be rounded brackets, square brackets, or double parentheses of the direct type. All the numbers in the matrix are called the matrix element, and they also have their coordinates in the field of the table. The matrix is necessarily indicated by a capital letter of the Latin alphabet.

The properties of matrices or mathematical tables include several aspects. The addition and subtraction of matrices is strictly element-wise. The multiplication and division of them goes beyond ordinary arithmetic. To multiply one matrix by another, we need to remember information about the scalar product of one vector on another.

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

The properties of matrix multiplication have some nuances. The product of one matrix by another is non-commutative, that is, (a, b) is not equal to (a, b).

The basic properties of matrices include such a thing as the measure of decency. The decency is considered a measure of decency for such tables. A determinant is a certain function of several elements of a square matrix entering the order n. In other words, a determinant is called a determinant. In a table with the second order, the determinant is equated to the difference of the products of numbers or elements of the two diagonals of this matrix A11A22-A12A21. The determinant for a matrix with a higher order is expressed by the determinants of its blocks.

To understand how degenerate the matrix, a notion such as the rank of the matrix was introduced. The rank is the number of independent linear columns and rows of the given table. The matrix can be invertible only if its rank is complete, that is, rank (A) is equal to N.

Properties of matrix determinants include:

1. For a square matrix, the determinant does not change when it is transposed. That is, the determinant of this matrix will be equated to the determinant of this table in a transposed form.

2. If any column or line contains only one zeros, then the determinant of such a matrix will be equated to zero.

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

4. If any column or any row of the matrix is multiplied by a number, then its determinant is multiplied by the same number.

5. If in the matrix any of the elements is written as the sum of two or more components, then the determinant of such a table is written as the sum of several determinants. Each determinant of such a sum is the determinant of the matrix, which instead of the element represented by the sum, one of the terms of this sum is written according to the order of the determinant.

6. If in any matrix there are two rows with the same elements or two identical columns, then the determinant of this table is equated to zero.

7. Also, the determinant is equal to zero for such a matrix, in which two columns or two lines are proportional to each other.

8. If the elements of a row or column are multiplied by a number, and then added to them by elements in another row or column of the same matrix, respectively, then the determinant of this table does not change.

In general, we can say that the properties of matrices represent a set of complex, but at the same time, necessary knowledge of the essence of such mathematical units. All properties of the matrix directly depend on its components and elements.