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Looking for a common solution of two systems

  1. Nov 29, 2012 #1
    Given two systems, Ax=b and Cy=d, for nxn matrices A and C, and n-dimensional vectors b and d, each of which has at least one solution, it is know that one solution is common to both (satisfies both equations). Could such solution be z found by solving Az+Cz=b+d? I understand that a common solution would satisfy the third system, but I wonder if the third system has more solutions, and we obtain one that is not common for the first and the second system.
     
  2. jcsd
  3. Dec 1, 2012 #2

    Erland

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    Science Advisor

    It certainly can!

    As an extreme example, let C=-A and d=-b, and assume that A is invertible. Then Ax=b has a unique solution, call it x0. This x0 is also a unique solution to By=d, so the systems have a unique common solution.

    But A+C is a zero matrix, and b+d is a zero vector, so all vectors z in Rn are solutions to (A+C)z=b+d.
     
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