Which of these statements are true?
(I) If $A$ is a matrix of full rank, then $A$ is invertible.
(II) If $A$ can be expressed as a product of elementary matrices, then $A$ is of full rank.
(III) If $A$ is symmetric, then a basis for the row space of $A$ is also a basis for the column space of $A$.
Understanding that equivalent statements are only for $A_{n \times n}$ matrix, so (I) cannot be true. I have put (II) and (III) to be the correct statements, am I correct?
Best Answer
(I) is, in fact, an "if and only if", so it's more than only true. You can find a proof here: Proof - Square Matrix has maximal rank if and only if it is invertible
(II) is also true. Note that $\text{det}(A\cdot B) = \text{det}A \cdot \text{det} B$. The determinant of a elementary matrix is never zero because they are invertible, so the product of determinants of the elementary matrices is not zero, implying that $\text{det}A \neq 0$, therefor invertible.
(III) holds trivially, but the reciprocal is not true.
As the statements are talking about invertibility and base span of a matrix you can assume that $A \in M_{n}$ (is a square matrix).