Final answer:
If a system of linear equations t AX=B has infinitely many solutions, it means the equations are not independent. The system AX=C can have zero, one, or infinitely many solutions, depending on the relationship of C to B and the original line/plane/space described by the equations.
Step-by-step explanation:
If a system of linear equations t AX=B has infinitely many solutions, this typically means that there is a dependency between the equations in the system, resulting in a situation where the equations are not independent. This usually happens when one equation is a multiple of the other, or both of them together describe the same line in the case of two variables, or plane/space in cases of three or more variables.
For the system AX=C with some other matrix C, it can have zero, one, or infinitely many solutions depending on how C relates to the original equations represented by B. If C falls on the same line/plane/space that B does, then the system AX=C will also have infinitely many solutions. If C does not align with that line/plane/space, the system will have no solutions because it represents a line/plane/space that does not intersect with the original. If the system of equations changes such that it becomes consistent and independent, then there will be a single solution.
Here's an example:
- Original system: x + y = 2 (equation A) and 2x + 2y = 4 (equation B which is just 2*A).
- Updated system: x + y = 3 (equation C).
- The original system has infinitely many solutions because equations A and B are dependent. Equation C, however, does not have the same solutions as A and B, so the new system (A and C together) either has no solution if the lines represented do not intersect, or exactly one solution if they do.