Eigenvalues and their algebraic multiplicities for a sum of two commuting matrices

Question

Question:

Let $A$ and $B$ be two real $n×n$ matrices which commute and let $a_1, a_2,..., a_n$ (not necessarily distinct) be the eigenvalues of $A$ and $b_1 ,...,b_n$ are eigenvalues (not necessarily distinct) of $B$.

(1) Are $a_1 + b_1,...,a_n +b_n$ the eigenvalues of $A+B$ ?

(2) If $a$ is eigenvalue of $A$ with algebraic multiplicity $k$ and $b$ is eigenvalue of $B$ with algebraic multiplicity $k$, then is $a+b$ an eigenvalue of $A+B$ with algebraic multiplicity $k$ ?

My attempt: I know that if $A$ & $B$ are commutating real matrices then they are simultaneously triangularizable & from this we can get, spectrum of $A+ B$ is contained in $\{a_1 + b_1 : a_1 ∈ σ(A) , b_1 ∈ σ(B) \}$ But then how can we get (1) and (2) from this?

Answer 1

Just consider $A=\pmatrix{1&0\\ 0&-1}$ and the following two different cases: $B=A$ and $B=\pmatrix{-1&0\\ 0&1}$.