If the row space and the column space of an n x n matrix A coinside, then matrix A is invertible.

Question

Hello guys i need help solving this problem. So this is my interpretation of the problem. I think its false because an orthogonal Matrix will have exactly the same row space and column space, however it is not invertible. The thing that confuses me is that do we need to show that is it invertible if it coinsides only (i mean that they are not equal) or both cases are the same.

Answer 1

Any orthogonal matrix $A$ is invertible; indeed, $A^{-1} = A^\top$. So one of your statements is definitely erroneous. However, if you take any symmetric $n\times n$ matrix of rank $<n$, it will fail to be invertible and, since $A^\top = A$, its row and column spaces coincide.