Finite-dimensional C*-algebra quotient

Question

Let $A$ be a finite-dimensional C*-algebra. Is it true that every quotient of $A$ is of the form $PAP$ for some projection $P\in A$?

(It's obviously true for commutative C*-algebras, that's why I am asking.)

Answer 1

Finite-dimensional C*-algebras are of the form $A = \oplus_1^n F_i$ where $F_i = M_{k_i}$ for some $k_i \in \mathbb{N}$. Its not too hard to convince yourself and that any ideal is of the form $I = \oplus_{i \in J} F_i$ where $J \subseteq \{1,\dots,n\}$. The quotient is then isomorphic to $A/I \simeq \oplus_{i \in \{1,\dots,n\} \setminus J} F_i$. From here, can you show that $A/I \simeq PAP$ for some projection $P$? (the same argument as finite-dimensional + commutative will work)