Does the equation upper a bold x equals bold b has at least one solution for every possible b?

Question:

Does the equation upper a bold x equals bold b has at least one solution for every possible b?

System of equation

For any matrix A and any vector X and b the system AX=b is called the system of equation. If the vector b=0 then the system is called the homogeneous system of equation otherwise non-homogeneous system of equation.

Let A be any matrix and X and b are the vectors then the system of equations AX=b has a solution if

Case(i) If {eq}b=0 {/eq}

Then there exists at least one solution i.e. Trivial solution will be the solution.

Case(ii) If {eq}b \neq 0 {/eq}

Then the system of the equation has either a unique solution if the rank of the augmented matrix A b is equal to the rank of the matrix A

and has no solution if the rank of the augmented matrix A b is not equal to the rank of the matrix A