Cramer's Rule

Cramer's Rule
Using Determinants to solve systems of equations

1. Vector Calculus And Linear Algebra Guided by : Dr. Ravi Tailor Patel Vinita .G. Presented by : Abdul Sattar (130940107004)

2. Cramer's Rule ( Using Determinants to solve systems of equations )

3. Introduction Suppose that we have a square system with n equation in the same number of variables ( ). Then the solution of the system has the following cases. 1) If the system has non-zero coefficient determinant D= det (A), then the system has unique solution and this solution is of the form where is the determinant obtained from D by replacing in D column by the column with the entries ( )

4. a) if at least one of D i is non-zero then the system has no solution b) if all D i ‘s are zero, then the system has infinite number of solutions. In this case, if the given system is homogeneous; that is, right hand side is zero then we have ffollowing possibilities of its solution. (1) if D i ≠ 0, then the system has only trival solution. (2) if D=0, then the system has also nontrival solutions. 2) If the system has zero coefficient determinant D= det (A) , then we have two possibilities as discussed below.

5. Example : 1 Find the solution of the system Solution : In matrix form, the given system of equations can be written as Ax=b, where

6. Here , matrix A is a square matrix of order 3, so Cramer’s rule can be applied Now, Therefore, the system has unique solution. For finding unique solution, let us first find D 1 ,D 2 and D 3 it can be easily verified that

7. Example : 2 Find the solution of the system Solution : In matrix form, the given system of equations can be written as Ax=b, where

8. Therefore, the system has unique solution.

9. Here , matrix A is a square matrix of order 3, so Cramer’s rule can be applied Now, Therefore, either the system has no solution or infinite number of solution. Let us check for it. Therefore, the system has no solution as at least one Di, i =1,2,3 is nonzero.

10. Example : 3 Find the solution of the system Solution : In matrix form, the given system of equations can be written as Ax=b, where

11. Here , matrix A is a square matrix of order 3, so Cramer’s rule can be applied Now, Also,

12. Therefore, the system has infinitle number of solutions. Now,

13. Therefore, þ(A)=2 Omitting m-r = 3-2 = 1 Considering n-r = 3-2 = 1 variable as arbitary , the remaining system becomes Where x is arbitary Now ,

14. Therefore , Let x = k , where k is arbitary , then the infinite number of solutions of the given system is where k is an arbitary constant.

15. Thank YOU !