Solve Unknown b in Matrix C for No Unique Solutions

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To determine values of b for which the matrix C leads to no unique solutions in the equation y = Cx, the rank of the matrix must be less than the number of variables. The discussion highlights that if b equals 3, the fourth column becomes a multiple of the first column, resulting in a rank deficiency. This condition indicates that the system of equations lacks a unique solution. The relationship between the determinant and the uniqueness of solutions is also noted, emphasizing that a non-invertible matrix correlates with non-unique solutions. Understanding the rank is crucial for solving the problem effectively.
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I have the following matrix C =

3 2 1 9
4 2 6 12
1 4 -3 3
0 1 8 (3-b)

y1=[-1 -1 1 -1] transpose

For the vector y, I need to find all values of b such that the system of equations y=Cx has no unique solutions. Can someone help...

Thanks
 
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What is the relation between the system determinant and the uniqueness of the solution?
 
I am not sure...I thought the system determinant is only used to find the interdependency of the basis. If there exist such a relation as you have mentioned...that is not mentioned in the question; so we can assume anything for the problem.
 
By the way in order to have a unique solution r(A), rank equals n. But we know n but I don't know how to get rank!
 
Notice how if b = 3 then column 4 becomes a multible of column 1. This solution would make the rank(C) < n = 4 and therefore C would not have an inverse making the system have no unique solitions.
 
Thanks Live2Learn...I got it!
 

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