Analyzing an unusual matrix

#linear_algebra

Let $a$ and $b$ be real numbers and let $A \in R^{3 \times 3}$ with each diagonal entry equal to $a$ and each off-diagonal entry equal to $b$ .

Determine all eigenvalues and representative eigenvectors of $A$ together with their algebraic multiplicities. (Hint: $A = (a - b) I + b J$ where $J$ is the $3 \times 3$ matrix each of whose entries equals $1$ .)
Is $A$ diagonalizable? Justify your answer.
Determine the minimal polynomial of $A$ .

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Let $a$ and $b$ be real numbers and let $A\in {\bf R}^{3\times 3}$ with each diagonal entry equal to $a$ and each off-diagonal entry equal to $b$.
\begin{enumerate}[label=\alph*)]
	\item Determine all eigenvalues and representative eigenvectors of $A$ together with their algebraic multiplicities.
	
	\medskip
	\noindent ({\itshape Hint:} $A=(a-b)I+bJ$ where $J$ is the $3\times 3$ matrix each of whose entries equals $1$.)
	
	\item Is $A$ diagonalizable? Justify your answer.
	\item Determine the minimal polynomial of $A$.
\end{enumerate}