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

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SUMMARY

The discussion focuses on solving the equation x = B-1(2A + I)(C-1 + A)b without computing any matrix inverses. Participants explore the transformation of the equation, specifically attempting to manipulate the terms to isolate C-1 effectively. The key insight involves rewriting the equation as Bx = (2A + I)C-1(I + CA)b, which allows for further simplification without direct inversion. The conversation highlights the importance of understanding matrix operations and manipulation techniques in linear algebra.

PREREQUISITES
  • Understanding of matrix operations, including multiplication and addition.
  • Familiarity with nonsingular matrices and their properties.
  • Knowledge of linear algebra concepts, particularly matrix inverses and transformations.
  • Experience with vector-matrix multiplication and its implications.
NEXT STEPS
  • Study techniques for matrix factorization to avoid direct inverses.
  • Learn about the properties of nonsingular matrices and their applications.
  • Explore advanced linear algebra topics, such as the Cayley-Hamilton theorem.
  • Investigate computational methods for solving linear systems without matrix inversion, such as LU decomposition.
USEFUL FOR

Mathematicians, students of linear algebra, and professionals in fields requiring matrix computations will benefit from this discussion, particularly those looking to enhance their skills in solving equations involving matrices without direct inverses.

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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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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...
 


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?
 

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