
#1
Nov2912, 11:31 AM

P: 87

Given two systems, Ax=b and Cy=d, for nxn matrices A and C, and ndimensional 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
Dec112, 12:21 PM

P: 304

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 x_{0}. This x_{0} 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 R^{n} are solutions to (A+C)z=b+d. 


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