I'm going through the topic of C$^*$-algebras and facing a few questions.
Q1. Consider the C$^*$-algebra $A=\bigoplus_{j=1}^n\mathbb C$ . What is the spectrum of $A$ (the collection of multiplicative non zero functionals) be?
I have tried to see $A$ as a direct sum but could not proceed further.
This is an important "change of frame" that might help you: $$\mathbb C^n\simeq C(\{1,\ldots,n\}).$$ From there you can immediately see that the spectrum of $A$ is the discrete topological space with $n$ points (or $\{1,\ldots,n\}$, if you want).
For a way less exciting alternative, you can prove directly that if $\phi:\mathbb C^n\to\mathbb C$ is linear and multiplicative, then it is one of the coordinate maps. Namely, being linear forces $$\phi(a_1,\ldots ,a_n)=\sum_{j=1}^n \beta_ja_j$$ for some $\beta_1,\ldots,\beta_n\in\mathbb C$. Now, since $\phi$ is multiplicative, $$ \beta_1+\beta_2=\phi(1,1,\ldots,0)=\phi((1,1,\ldots,0)^2)=\phi(1,1,\ldots,0)^2=(\beta_1+\beta_2)^2, $$ from where you get that either $\beta_1=0$ or $\beta_2=0$. Now do the same between the surviving one and $\beta_3$, etc., to show that only one of the $\beta_j$ is nonzero.