| # Full Rank Matrices Theorem A matrix is full column rank if and only if is invertible. full row rank if and only if is invertible. Proof: The matrix is full column rank if and only if its nullspace if reduced to the singleton , that is, If is invertible, then indeed the condition implies , which in turn implies . Conversely, assume that the matrix is full column rank, and let be such that . We then have , which means . Since is full column rank, we obtain , as desired. The proof for the other property follows similar lines.