Existence of a Solution for Matrix Equations

December 2, 2010 by · Leave a Comment
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Let A be an m \times n matrix, x is n \times 1 and b is m \times 1

Ax = b\\ rank(A) = r\\ rank(\left[ A b \right] ) = r_a

If r_a = r < n then there are an infinite number of solutions.
If r_a = r = n then there is one unique solution.
If r < r_a then there are no solutions.

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