Determinants and their properties. Matrix. Operations with matrices. The inverse matrix. Systems of linear equations
1. Determinants and their properties
3. Nonsingular matrices
4. Systems of linear equations
Determinants and their properties
Determinants of second and third order
Consider the system of two linear algebraic equations in two variables
where a11,a12,a21, and a22 are coefficients, b1 and b2 are right-hand sides, and x1 and x2 are unknowns.
Let us solve this system by the school method of algebraic addition, namely, multiply the first and second equations by à22 and – a12 , respectively, and sum the results. The coefficient of õ2 will vanish. The remaining unknown, õ1, is found as follows:
In a similar manner, multiplying the first equation of the system by – à21, the second by à11 , and summing the resulting equations term by term, we obtain
Definition. The number
is called a determinant of second order.
The numbers a11,a12,a21 and a22 are called the elements of the determinant.
The second-order determinant is equal to the product of the elements of the main diagonal minus the product of the elements of the secondary diagonal.
Example . Find the determinants
The unknowns õ1 and õ2 of the linear system (1) are determined by the formulas
The determinant is called the principal determinant of the system; it is formed by the coefficients of the unknowns. x1 and x2are auxiliary determinants; they are obtained by replacing the elements of the first and second columns by the free terms of system (1).
A third-order determinant is the number
The simplest method for calculating a third-order determinant is the triangle rule.
The main diagonal of the determinant is the line containing the elements a11,a22 and a33.
The secondary diagonal is the line containing the elements a13,a22 and a31.
The products of the main diagonal elements and of the elements contained in the triangles shown below are summed with the plus sign:
The products of the secondary diagonal elements and of the elements contained in the triangle shown below are summed with the minus sign:
Another method for calculating a third-order determinant is as follows. We write the first columns on the right of the determinant, and sum the products of the elements of the main diagonal and of the two parallel diagonals with the plus sign. Then we add the products of the elements of the secondary diagonal and of the two parallel diagonals with the minus sign
+ + +
– – –
As a result, we obtain the determinant.
Example. Calculate the determinant by the triangle rule:
The determinant of order n is the expression
An nth-order determinant contains n2 elements.
The subscript, i, indicates the number of the row and the second subscript j, indicates the number of the column containing the element àij.
Properties of determinants. All determinants of any order have the same properties. For simplicity, we give only properties of third-order determinants.
1. The interchange of rows and columns in a determinant does not change its value:
= (the transposed determinant).
2. The interchange of two rows (columns) in a determinant changes only the sign of the determinant:
3. If all elements of any lines (a row or a column) are zero, then the determinant is equal to zero:
The proof follows from the triangle rule.
4. A determinant containing two equal lines is equal to zero:
5. The common multiplier of all elements of a line can be factored out:
6. A determinant containing two proportional lines, is equal to zero:
7. If each element of some line is the sum of two terms, then such a determinant equals the sum of two determinants, which contain these terms instead of the elements of the lines.
8. A determinant does not change under the replacement of any line by the sum of this line and any parallel line multiplied by some number.
Algebraic complements and minors.
Definition. The minor of an element àij is the determinant of order lower by one consisting of the elements that remain after the deletion of the ith-series and jth-column, which intersect in aij.
For example, the minor of the element a32 is
is the minor of .
Definition. The algebraic complement of an element àij is the minor of aij multiplied by -1 raised to the proper equal to the sum of the numbers of the row and the column intersecting in the given element:
9. A determinant equals the sum of products of all elements of any lines and the corresponding algebraic complements.
For a kthorderdeterminant, we can write property 9 in the form of expansion along the kth-column:
10. The sum of the products of the elements of any line and the algebraic complements of the corresponding elements of a parallel line equals zero:
Examples. Let us expand the following determinant along the third row: