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

[humantripod]

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

 

[obafgkmrns]

"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?

 

[humantripod]

Only 1. So does this mean there is no solution for Ax=c?

 

[obafgkmrns]

It means there is only one solution for Ax=c:

Inv(A)A x=Inv(A)c → x=Inv(A)c

 

[AlephZero]

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.