The discussion revolves around a matrix problem involving the existence of a matrix B that satisfies certain equations with a non-singular matrix A. Participants are examining the differences between two parts of the problem, specifically focusing on the implications of the wording and the conditions under which the equations hold.
Some participants have provided insights into the differences between part a and part b, suggesting that part b's requirement for a universal solution is problematic. There is an ongoing exploration of how to articulate the reasoning behind the impossibility of finding such a matrix B.
Participants are grappling with the implications of the term "any" in the context of the problem, which raises questions about the universality of the proposed solution. There is also mention of the need to address parts c and d, indicating that the discussion is still evolving.
HallsofIvy said:Part b is quite different from part a- and the difference is important to learn. Mathematics must be very very precise in its wording- unlike science we don't have observations and experiments to fall back on. In other words, we can't just look at the real world- words are everything!
In part a it ask if, given a non-singular matrix A, there exist a matrix B such that [itex]AB^2= A[/itex]. You are right- just multiply, on the left, on both sides by [itex]A^{-1}[/itex], which exists because A is non-singular, and the equation becomes AB= I. Yes, B exists and is the inverse of A.
In part B, it asks if there exists a matrix B such that, for any non-singular matrix, A, [itex]A^2B= A[/itex]. "Any" is the crucial word there. Is there a single matrix B that is the inverse of all invertible matrices?
HallsofIvy said:"No, there does not exist a single matrix, B, such that [itex]A^2B= A[/itex] for all non-singular matrices, A."