Math. Dept., Wuhan University of TechnologySec.4 Cramers Rule(克拉默法则克拉默法则) 1. System of Linear Equations2.Cramers Rule3. Review and QuestionsMath. Dept., Wuhan University of Technology1. System of Linear Equations(线性方程组）线性方程组）Linear Equation:System of linear equations: n unknowns, m equations(1)Math. Dept., Wuhan University of TechnologyWhen m=n, that is, n unknowns（未知量）（未知量）, n equations, The Coefficient Determinant（系（系数行列式）数行列式） of the linear system.Math. Dept., Wuhan University of TechnologythenIf 2.Cramers Rule Theorem 1Then the LS has a unique solution which are got as : Math. Dept., Wuhan University of TechnologyProofIf (1) has solution,then Math. Dept., Wuhan University of TechnologyMath. Dept., Wuhan University of TechnologyHence we have Math. Dept., Wuhan University of TechnologyExample 1 Solve the linear equations using Cramers rule:Solution.The Coefficient Determinant of this LS is:Math. Dept., Wuhan University of Technologyso Math. Dept., Wuhan University of TechnologyWhen the right-hand side members are all zero ,(1) becomeWhen arent all zero,The LS is called non-homogenous system of linear equations. (非齐次线性方程组非齐次线性方程组)(2)Homogenous System of Linear Equations （齐次线性方程组）（齐次线性方程组）Math. Dept., Wuhan University of TechnologyTheorem 2. If the coefficient determinant of a homogeneous linear equationthen the system has only one zero solution,Has only zero solutionHas many nonzero solutionsIn another word,If a homogeneous linear equation has nontrivial solutions(非零解非零解),then the coefficient determinant of the systemWe call it the trivial solution(零解零解)Math. Dept., Wuhan University of TechnologySo whenthe system has nontrivial solutions.Solution: If the system has nontrivial solution, the coefficient determinant must be zero.Math. Dept., Wuhan University of TechnologyReview1.Under what condition can we use the Cramers rule?2. The formula of the Cramers ruleMath. Dept., Wuhan University of TechnologyQuestion :Please find the linear equation which passes point Solution: Suppose the linear equation is There must be Where a,b,c cannot be all zero.So we have:Taking a,b,c as unknowns, it means (1) has nontrival solution. So the coefficient determinant of (1) is 0, it follows that(2) is a linear equation.Since satisfy (2) (2) Must be the linear equation we need.