How can I determine all the matrix A 3x3 in R such as AB=BA for all B 3x3 in R (without determinants)

Question

Determine all the matrix A 3x3 in R such as AB=BA for all B 3x3 in R (without determinants)

Answer 1

Let $e_k$ be the vector of zeroes with one in the $k$th position.

Then $Ae_ie_j^T = e_i e_j^T A$ and so $e_a^T Ae_ie_j^T e_b = e_a^T e_i e_j^T A e_b$, or more simply $[A]_{ai} \delta_{jb} = [A]_{jb} \delta_{ai}$.

Setting $a=i, j=b$ gives $[A]_{ii} = [A]_{jj}$ and setting $a=i$ and $j \neq b$ gives $0=[A]_{jb}$.

Hence $A$ must be in the span of the identity.