For what values of $\ a $ $\ \dim(\ker T) \oplus \dim(\operatorname{Im}T) = \mathbf R^3 $

Question

Let $\ A $ represent a transformation from $\ \mathbb R^3 \rightarrow \mathbb R^3 $

$$\ \begin{bmatrix} 1 & 1 & a+1 \\ 1 & a+1 & 1 \\ a+1 & 1 & 1 \end{bmatrix} | \ a \in \mathbf R$$

Find for which values of $\ a $ $\ \ker T \oplus \operatorname{Im}T = \mathbf R^3 $

Is it true that it is a direct sum whenever $\ T$ is isomorphism ?

Also is it true that $\ (\ker T) \oplus (\operatorname{Im}T) \not = \mathbf R^3 $ only if $\ \ker T \subseteq \operatorname{Im}T $ ?

Answer 1

1. Yes, but that's not all the cases.
2. No. Consider nonzero $\mathcal T$ with $\mathcal T^2 = \mathcal O$, then $\mathrm {Im}(\mathcal T) \subseteq \mathrm {Ker}(\mathcal T)$. In this case, $\mathrm {Ker}(\mathcal T) \subseteq \mathrm {Im}(\mathcal T)$ cannot happen according to the nullity-rank theorem.