Matrix Theory 1.5 Cramer Theorem

Section 5 Cramer Theorem

Set the linear equations composed of N Linear Equations containing n unknown numbers

(I)

Composed of coefficients of unknown numbersNOrder Determinant

CalledNCoefficient determinant of the linear equations (I.

Theorem 3 (Cramer theorem) if the coefficient determinant of the linear equations (I) is not equal to zero

,

Then, equations (I) have a unique solution, and the solution can be expressed as a determinant.

, ××× ,,

WhereDJ(J= 1, 2, ×××,N) Is to determine the CoefficientDCenterJColumn elementA1j,A2j, ×××,AnJ is equivalent to the constant of the equations.B1,B2, ×××,BNTheNOrder Determinant, that is

. (J = 1, 2 ,..., N)

Note: In the Cramer theorem,

(1) d = 0;

(2) The solution is unique;

(3) The only solution is

Therefore, when solving the linear equations (I), first obtain the coefficient determining factor D. When D is less than 0, then obtain the other N determining factors DJ (j = 1, 2 ,..., N), then the unique solution of the equations (I) is given.

Example 17 Solving Linear Equations

.

Solution

,

,

,

,

That's why

,,,.

The inverse negative proposition of the Cramer theorem is:

Theorem 4 linear equations (I) without solution or solution is not unique, the coefficient determinant D = 0

When the constant term of the linear equations (I)B1 =B2 = ××× =BNWhen the value is 0, the linear equations (I) are

(Ii)

It is called the homogeneous linear equations of n elements. Correspondingly, the constant term on the right side of the linear equations (I)B1,B2, ×××,BNWhen the sum is zero, the linear equations (I) are called non-homogeneous linear equations of n elements.

Theorem 5 if the coefficient determinant of homogeneous linear equations (ⅱ)DIf limit 0 is used, the homogeneous linear equations (ⅱ) have only zero solutions (no non-zero solutions ).

Note: Linear Equations (ⅱ), whether or not D is zero, have zero solutions (the solution is zero),DWhen limit 0 is used, there is only one zero solution. When D is set to 0, there are other solutions except zero solution. This issue will be discussed later.

Example 18 Homogeneous Linear Equations

There is only zero solution. Evaluate the value of λ.

Solution: coefficient Determinant

.