# Existence of a Solution for Matrix Equations

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Let be an matrix, is and is

If then there are an infinite number of solutions.

If then there is one unique solution.

If then there are no solutions.

Therefore, when is a linear combination of the columns in , then (since is not independent of the other columns in , it will not add to the rank) and there is at least one solution. But if is not a linear combination of the columns in then there are no solutions.