The properties of matrices is a question that can cause difficulties for many. Therefore, it is worth considering it in more detail.
A matrix is a rectangular table that includes numbers and elements. It is also a collection of numbers and elements of some other structure, which are written as a rectangular table, consisting of a certain number of rows and columns. Such a table must necessarily be enclosed in brackets. It can be round brackets, square brackets or double brackets of direct type. All numbers in the matrix are called a matrix element, and they also have their coordinates in the table field. The matrix is mandatory indicated by the capital letter of the Latin alphabet.
The properties of matrices or mathematical tables include several aspects. Addition and subtraction of matrices is strictly element-wise. Multiplying and dividing them is beyond the scope of ordinary arithmetic. To multiply one matrix by another, you need to recall information about the scalar product of one vector by 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 a measure of decency. Decency is considered a decency measure for such tables. A determinant is a certain function of several elements of a square matrix in order n. In other words, the determinant is called the determinant. In a table with a second order, the determinant is equal to the difference of the products of numbers or elements of 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 concept such as the rank of the matrix was introduced. A rank is the number of linearly independent columns and rows in a given table. A matrix can be invertible only when its rank is complete, i.e. rank (A) is N.
The properties of matrix determinants include:
1. For a square matrix, the determinant will 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 row will include only zeros, then the determinant of such a matrix will be equal to zero.
3. If in the matrix any two columns or any two rows are interchanged, then 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 any number, then its determinant is multiplied by the same number.
5. If any of the elements is written in the matrix as the sum of two or more components, then the determinant of such a table is recorded as the sum of several determinants. Each determinant of such a sum is a determinant of a matrix in 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 identical elements or two identical columns, then the determinant of this table is equal to zero.
7. Also, the determinant is equal to zero for such a matrix in which two columns or two rows are proportional to each other.
8. If the elements of a row or column are multiplied by a number, and then the elements in another row or column of the same matrix are added to them, respectively, then the determinant of this table will not change.
In total, we can say that the properties of matrices are a set of complex, but at the same time, necessary knowledge about the essence of such mathematical units. All the properties of a matrix directly depend on its components and elements.