Cramers Rule and unique solutions

  • Thread starter nick484
  • Start date
In summary, the conversation discusses finding values for s and t that result in a unique solution for a given system of equations. While s=2 and t=1 do not have a unique solution, there are other possible values such as s=4 and t=2 that do. The conversation also mentions using Cramer's rule to find solutions, relying on the determinant of the coefficient matrix being non-zero.
  • #1
nick484
2
0
Hey

Im having trouble of how to go about this. Afterwards we have to perform some Cramers rule operations however I am currently struggling. For:

Find all values of s and t for which the following system has a unique solution:
sx1 + 2x2 + x3 + 3x4 = 1
tx1 + x2 + 3tx3 + s^2x4 = 0
sx3 + 2tx4 = 2
x3 + x4 = 1

Do I just say that (s/2)=/1 and t=/1 because of the last two equations or is there more to it?

Thanks
 
Physics news on Phys.org
  • #2
Well, s= 2, t= 1 will cause there to be no solution but those are not the only values. For example, if s= 4 and t= 2, the final two equations wil be 4x3+ 4x4= 2 which reduces to x3+ x4= 1/2, contradicting the last equation.

Since you talk about Cramer's rule, I assume you know that you can write the solutions to such a system as one determinant divided by the other- and you can do that as long as the denominator, the determinant of the coefficient matrix, is non-zero.
 
Last edited by a moderator:
  • #3
Thanks
 

FAQ: Cramers Rule and unique solutions

1. What is Cramer's Rule?

Cramer's Rule is a method used to solve systems of linear equations by using determinants. It allows us to find unique solutions to systems that have the same number of equations and unknown variables.

2. How does Cramer's Rule work?

Cramer's Rule involves finding the determinants of the coefficient matrix and the constant matrix in a system of linear equations. These determinants are then used to calculate the values of each variable in the system, resulting in a unique solution.

3. What are the requirements for using Cramer's Rule?

Cramer's Rule can only be used for systems of linear equations that have the same number of equations and unknown variables. The coefficient matrix must also have a non-zero determinant in order for Cramer's Rule to be applicable.

4. Can Cramer's Rule always find a unique solution?

No, there are cases where Cramer's Rule may not be able to find a unique solution. This can happen if the coefficient matrix has a determinant of zero or if there are infinitely many solutions to the system of equations.

5. Are there any alternatives to using Cramer's Rule?

Yes, there are other methods for solving systems of linear equations such as elimination, substitution, and matrix inversion. These methods may be more efficient or easier to use depending on the specific system of equations.

Similar threads

MHB Find an A and b such that Ax=b has the solution set......
Replies
4
Views
2K
What kind of form is this general solution of a system?
Replies
1
Views
1K
Express the system in state space form
Replies
2
Views
1K
MHB Maximizing Tr(A) & Unique Solution of Matrix A w/ Infinite Solutions
Replies
1
Views
1K
Nullspace Matrix Homework: Struggling to Find an Example Vector
Replies
7
Views
1K
MHB Systems of equations - further understanding
Replies
2
Views
1K
MHB Is $s$ the unique vector that spans the solution space $L(A,0)$?
Replies
3
Views
1K
Describe every solution to Ax=0
Replies
2
Views
3K
Working with differential equations to obtain a function
Replies
8
Views
3K
Calculate T for Maximum Altitude: 1D Kinematics Problem Solution
Replies
5
Views
3K

Hot Threads

Recent Insights

Back
Top