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Matrices: Number of solutions of Ax=c if we know the solutions to Ax=b

  1. Jan 12, 2012 #1
    Hey guys,

    Here is my question.

    A is a 4x4 matrix and there are two vectors, b and c, which have 4 real numbers. If we are told that A(vector x)=(vector b) has an unique solution, how many solutions does A(vector x)=(vector c) have?

    I honestly have no idea how to do this. I know that for A would be in the following rref form:

    1 0 0 0
    0 1 0 0 for Ax=b.
    0 0 1 0
    0 0 0 1
     
  2. jcsd
  3. Jan 12, 2012 #2
    "A(vector x)=(vector b) has an unique solution" is another way of saying that A has an inverse. How many inverses can a matrix have?
     
  4. Jan 12, 2012 #3
    Only 1. So does this mean there is no solution for Ax=c?
     
  5. Jan 12, 2012 #4
    It means there is only one solution for Ax=c:

    Inv(A)A x=Inv(A)c → x=Inv(A)c
     
  6. Jan 12, 2012 #5

    AlephZero

    User Avatar
    Science Advisor
    Homework Helper

    Another way to see it: suppose there is more than one solution to Ax = c.

    If Ax1 = c and A x2 = c, then A(x1-x2) = 0

    So if Ax = b, would be another solution A(x+x1-x2) = b

    But there is only one solution to Ax = b.
     
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