Solving the Formula Without Matrix Inverses: A,B,C & b

In summary: If so, then the proposed solution, Bx=(2A+1)C-1(I+CA)b, is correct. This is because if we multiply both sides of the original equation by C, we get:CBx = B-1(2A+I)(I+CA)b = (2CAC-1+2CA2+I+A)bSince C-1 is nonsingular, we can rearrange the terms to get:CBx = (I+2CAC-1+2CA2+A)b = (2CAC-1+2CA2+I+A)bTherefore, the proposed solution is not only correct, but it also avoids computing any matrix inverse. In summary, to implement the formula
  • #1
Euphz
1
0
If A,B and C are nxn matrice, with B and C nonsingular, and b is an n-vector, how would you implement the formula



x = B-1(2A+I)(C-1+A)b, without computing any matrix inverse?



I made it Bx=(2A+1)C-1(I+CA)b, but don't know how to pull out the C-1 from the middle in order to take the equation without matrices inverse,sigh.. but I don't know if I approached it right because I don't even get the question.

Please help me!
Thank you very much :frown:
 
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  • #2


The furthest I could come was by working out the parentheses:

CBx = (2CAC-1+2CA2+I+A)b

I hope someone else can get some answer...
 
  • #3


Euphz said:
If A,B and C are nxn matrice, with B and C nonsingular, and b is an n-vector, how would you implement the formula



x = B-1(2A+I)(C-1+A)b, without computing any matrix inverse?



I made it Bx=(2A+1)C-1(I+CA)b, but don't know how to pull out the C-1 from the middle in order to take the equation without matrices inverse,sigh.. but I don't know if I approached it right because I don't even get the question.

Please help me!
Thank you very much :frown:
What do you mean "implement the formula"? Is the goal to evaluate the right hand side of the original equation to get x?
 

1. What is the purpose of solving the formula without using matrix inverses?

The purpose of solving the formula without using matrix inverses is to simplify the process and make it more efficient. Matrix inverses can be time-consuming and computationally expensive, so finding alternative methods can save time and resources.

2. How is the formula solved without using matrix inverses?

The formula can be solved by using methods such as Gaussian elimination, Cramer's rule, or LU decomposition. These methods involve manipulating the matrix equations to reduce them to simpler forms that can be solved using basic algebra.

3. What are the advantages of solving the formula without matrix inverses?

Solving the formula without matrix inverses can be advantageous because it can be faster and more efficient, especially for larger matrices. It also allows for more flexibility in choosing the method of solving, as different methods may be more suitable for certain types of equations.

4. Are there any limitations to solving the formula without matrix inverses?

Yes, there are some limitations to solving the formula without matrix inverses. Some methods may only work for certain types of equations or matrices, and they may not always provide an exact solution. Additionally, the accuracy of the solution may depend on the precision of the calculations.

5. How does solving the formula without matrix inverses relate to real-world applications?

Solving the formula without matrix inverses has many practical applications, especially in fields such as engineering and science. It can be used to solve systems of linear equations, which are commonly used to model and solve real-world problems. By finding alternative methods to solve these equations, we can improve the efficiency and accuracy of our solutions.

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