Relation of a Matrix and a System of Equations Question

[Liquid7800]

Hello, I just have a general question regarding if my analysis of the relationship of system of equations and the matrices constructed involved in solving them.

If systems of equations can be solved through matrices (Vandemonde...etc) by obtaining the inverse of the coefficient/variable matrix, then could it be said that if some of the variables don't satisfy the system of equations ...does that mean that an inverse of the matrix (of the system of equations in question) will not exist?

In other words, if the system of equations can't be solved..then the inverse of the matrix (from the system of equations) will also correspondingly not exist.

Thanks, appreciate any info

 

[adriank]

In the equation Ax = b, the inverse of A exists if and only if there is exactly one solution for every b.

 

[Liquid7800]

If I try to find the determinant of a matrix of equations such that Ax = b,and not every 'b' has a solution will I return 0?

Thanks...

 

[HallsofIvy]

Yes. If the determinant of A were not 0, then A would have an inverse and we would have $x= A^{-1}b$ as the unique solution for any vector, b. If there exist a vector, b, for which Ax= b does not have a solution, then $A^{-1}$ cannot exist and the determinant of A must be 0.

(I am assuming here a finite system of equations.)

 

[Liquid7800]

Thank you for the explanation--- I appreciate the well stated answer and it makes perfect sense to me now.