Let A be a real 3times 3 matrix. Which of the following conditions does not imply that A is invertible?

Question

Let A be a real $3\times3$ matrix which of the following conditions does not imply that A is invertible?

(D) The set of all vectors of the form $Av$, where $v \in \mathbb R^3$, is $\mathbb R^3$.

(E) There exist 3 linearly independent vectors $v_1$, $v_2$, $v_3 \in \mathbb R^3$ such that $Av_i\neq 0$ for each $i$.

The answer is E. I see why E is the answer by coming up with a counter example. But why does D imply that A is invertible? Also, how to solve this problem in a more efficient way (instead of coming up with a counter example)?

Answer 1

$A$ is invertible iff the rank is 3. The rank is the dimension of the image. So D) says that the rank is $3$. For E) there may be three vectors with nonzero image, but they may all be sent to the same vector so it doesnt guarantee that the image has dimension $3$.

Answer 2

Let $e_1 = (1,0,0)^T, e_2 = (0,1,0)^T, e_3 = (0,0,1)^T$, D implies the existence of $v_1, v_2, v_3$ such that $Av_1 = e_2, Av_2 = e_2, Av_3=e_3.$ If $V = \left[ \begin{matrix} v_1 & v_2 & v_3 \end{matrix} \right]$ then $AV = I$ which means $A$ is invertible and $V$ is the inverse.