Currently I'm reading Rørdam, Larsen and Laustsen's book "An introduction to $K$-theory for $C^*$-algebras" and I'm stuck to proof Lemma 4.3.1. (ii). The statement is:
Let $$0\to J \xrightarrow{\varphi} A\xrightarrow{\psi} B\to 0$$ be ahsort exact sequence of $C^*$-algebras with $\varphi$ and $\psi$ *-homomorphisms. Let $n\in\mathbb{N}$.
(i) The mapping $\varphi_n^+:M_n(J^+)\to M_n(A^+)$ is injective.
(ii)An element $a\in M_n(A^+)$ belongs to the image of $\varphi_n^+$ if and only if $\psi_n^+(a)=s_n(\psi_n^+(a))$.
The notations means: $J^+$ means the unitization of $J$, $\varphi^+$ is the canonical extension of $\varphi$ on the unitizations and $\varphi_n$ is the induced map of $\varphi$ on the matrices, i.e. a map $M_n(J)\to M_n(A)$. And $s$ is the scalar mapping, which sends an element in the unitization of it's scalar-part.
Part (i) is clear and therefore it is $M_n(J^+)\cong im(\varphi_n^+)$. To part (ii):
At first to the idea of $\Rightarrow$: Given $a\in M_n(A^+)$ such that $a\in \operatorname{im}(\varphi_n^+)$. Write $\operatorname{im}(\varphi_n^+)=\varphi_n^+(M_n(J^+))=\varphi_n(M_n(J))+\varphi_n^+(M_n(\mathbb{C}1_{J^+}))=\operatorname{im}(\varphi_n)+\varphi_n^+(M_n(\mathbb{C}1_{J^+}))$. Furthermore, by exactness it is $\operatorname{im}(\varphi_n)=\operatorname{ker}(\psi_n)$. Then $\psi_n^+$ vanishes on the part of $a$ which lies in $\operatorname{im}(\varphi_n)$ and for the part of $a$ which lies in $\varphi_n^+(M_n(\mathbb{C}1_{J^+}))$ (this part is scalar), we have $\psi_n^+(a)=s_n(\psi_n^+(a))$.
And the Direction $\Leftarrow $ is more or less the same argument backwards.
I want to imrove proof formally and my problem is: Are these equalities $\varphi_n(M_n(J))+\varphi_n^+(M_n(\mathbb{C}1_{J^+}))=\operatorname{im}(\varphi_n)+\varphi_n^+(M_n(\mathbb{C}1_{J^+}))$ correct and can I write $a\in \operatorname{im}(\varphi_n^+)$ as a sum $a=a_1+a_2$ with $a_1\in \operatorname{im}(\varphi_n)$ and $a_2\in \varphi_n^+(M_n(\mathbb{C}1_{J^+}))$?
By identifying $\mathbb{C}1_X$ with $\mathbb{C}$ to simplify the notation you indeed have $M_n(X^+) = M_n(X) \oplus M_n(\mathbb{C})$ (as vector spaces). The maps $(\phi^+)_n$ and $(\psi^+)_n$ are induced by $\phi$ and $\psi$ on the first component and by the identity on the second component. $s$ is simply the projection onto the second component. With this description it should be easy to verify that your proof indeed works.