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### A be a $3\times 3$ matrix over $\mathbb {R}$ such that $AB =BA$ for all matrices $B$. what can we say about such matrix $A$ [duplicate]

Let $A$ be a $3\times 3$ matrix over $\mathbb {R}$ such that $AB =BA$ for all matrices $B$ over $\mathbb {R}$ then what can we say about such matrix $A$. or such matrix $A$ must be orthogonal matrix? ...
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### Centre of a matrix ring are $\operatorname{diag}\{ a, a, …, a \}$ with $a\in Z(R)$ [duplicate]

Show that $Z(M_n(R))$ consist of $\operatorname{diag}\{ a, a, ..., a \}$ with $a\in Z(R)$
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### Matrices that commute with all matrices [duplicate]

Let $Z_n$ be the set of all $n \times n$ matrices that commute with all $n \times n$ matrices. Show that $$Z_n = \{\lambda I_n \ | \ \lambda \in \mathbb R\}$$ ($I_n$ is the $n \times n$ identity ...
### proof that if $AB=BA$ matrix $A$ must be $\lambda E$ [duplicate]
Let $A \in Mat(2\times 2, \mathbb{Q})$ be a matrix with $AB = BA$ for all matrices $B \in Mat(2\times 2, \mathbb{Q})$. Show that there exists a $\lambda \in \mathbb{Q}$ so that $A = \lambda E_2$. ...