Basic Matrix Theoretical Information

What is the Matrix?

A matrix is a table filled with numbers. The most important characteristics of a matrix are the number of rows and the number of columns. A matrix with the same rows and columns makes a square.

The numbers themselves are called matrix elements and characterize their position in the matrix, specifying the row number and the column number and writing them as a double index, and first, the row number and then the column number is written. For example, a 14 is the matrix element in the first row and the fourth column, a 32 is in the third row and second column.

The main diagonal of a square matrix is called the elements that have the same indices, that is, those elements whose row number coincides with the column number. The side diagonal runs “perpendicular” to the main diagonal.

Of particular importance are the so-called identity matrices. In a square matrix on the main diagonal with 1 and equal to zero to all other numbers. They denote the identity matrices E. Matrices are called equal if they have the same number of rows, the number of columns, and all elements with the same indices is equal. A matrix is ​​called zero if all its elements are equal to 0. The zero matrices A is denoted. The matrix multiplication calculator makes your calculation easy.

The simplest operations with matrices -

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

2. Addition of matrices. Only matrices of the same size, that is, having the same number of rows and the same number of columns, can be added. When matrices are added, their corresponding elements are added.

3. Transposition of the matrix. When the matrix is ​​transposed, the rows become columns and vice versa. The resulting matrix is called transposed and denoted A T.

Matrix transposition properties

4. Matrix multiplication.

The following properties exist for the matrix product -

If the number of columns in the first matrix is ​​equal to the number of rows in the second matrix, then the multiplication of two matrices is possible.

And we will get a result matrix whose number of rows is equal to the number of rows of the first matrix, and the number of columns is equal to the second matrix’s number of columns.

Matrix multiplication is noncommutative. This means that the result changes from the permutation of the matrices in the product. Moreover, if you can count the product A ∙ B, this does not mean at all that you can count product B ∙ A.

Let C = A ∙ B. To determine the element of the matrix C, standing in the i -th row and k -th column, it is necessary to take i -th row of the first matrix to be multiplied and the k -th column of the second. We take the first element from the row of the first matrix and multiply it by the first element of the column of the second matrix.

Matrix product properties

• Determinant of a matrix

The determinant (determinant) of a square matrix A is a number, which is denoted by det A, less often | A | or just Δ, and is calculated in a certain way. For a 1x1 matrix, the determinant is the single element of the matrix itself.

Determinant of a 2x2 matrix

Minors and Algebraic Complements

Consider the matrix A. Choose s rows and s columns in it. Let’s compose a square matrix of elements at the intersection of the resulting rows and columns. The minor of matrix A of order s is the determinant of the resulting matrix.

Consider a square matrix A. Choose s rows and s columns in it. An additional minor to a minor of order s is called a determinant composed of the elements remaining after deleting these rows and columns.

The algebraic complement to the element a ik of the square matrix A is the additional minor to this element, multiplied by (–1) i + k. Denote the algebraic complement A ik.

Calculation of the determinant of a matrix in terms of algebraic complements

Consider a square matrix A. If you want to calculate a determinant, you need to select any of its rows or columns and find the products of each element of this row or column by the algebraic complement to it. And then all these works must be summed up.

When calculating algebraic complements, do not forget about the factor (–1) i + k. To make the counting easier, select the row or column of the matrix that contains the largest number of zeros.

Calculation of the algebraic complement can be reduced to the calculation of a determinant with a size of more than 2x2. In this case, such a calculation also needs to be carried out through algebraic additions, and so on until the algebraic additions that need to be counted become 2x2, then use the formula above.