# Linked Questions

4answers
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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? ...
1answer
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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 $a\in Z(R)$
2answers
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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 ...
2answers
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### 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 $A = \lambda E_2$. ...
2answers
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Given a (not necessarily finite dimensional) vector space $V$ prove that the center of $\operatorname{GL}(V)$ is the set of all scalar transformations (i.e all transformations of the form $a\... 6answers 382 views ### Let$A$be a$3 \times 3$matrix with real entries such that… [duplicate] I came across the following problem that says: Let$A$be a$3 \times 3$matrix with real entries.If$A$commutes with all$3 \times 3$matrices with real entries,then the number of distinct ... 3answers 379 views ### Let$A$be an$n\times n$matrix over real numbers such that$AB=BA,$for all$n\times n$matrices$B$[duplicate] I was thinking about the following problem: Let$A$be an$n\times n$matrix over real numbers such that$AB=BA,$for all$n\times n$matrices$B$.Then which of the following options is correct? ... 2answers 593 views ### A linear operator$T: V \rightarrow V$commuting with all linear operators is a scalar multiple of the identity. [duplicate] Let$\mathbb{K}$a field,$V$a vector space over$\mathbb{K}$. If$T:V\to V$commutes with all other linear operators$V \to V$, then there exists$\lambda \in \mathbb{K}$such that$T= \lambda I$, ... 0answers 1k views ### The Center of a Matrix Ring [duplicate] Prove that the center of the ring$M_n(R)$is the set of scalar matrices. I know what a center look like but i feel like i have not enough information to even solve this problem. Anyone that can help ... 0answers 604 views ### If$f$commutes with every linear transformation, it's a scalar multiple of the identity [duplicate] I'm dealing with a problem related to linear transformation. Problem: Let$f \in L\left( V \right)$, where$L\left( V \right)$is the set of all linear operators on$V$. Prove that if$fg = gf$... 1answer 367 views ### If$AB = BA$, then$A= λI_n$? [duplicate] Prove that :: If$A$is an$n\times n$matrix such that$AB = BA$for any$n\times n$matrix$B$, then$A=\lambda I_n$2answers 103 views ### Find all$3\times3$square matrices which commute with any$3\times3$upper triangular matrix. [duplicate] I'm not sure how to proceed. Let us find all possible solutions for the matrix$A$which commutes with any other matrix$X$. In other words: $$AX=XA$$ Stating the matrix multiplication explicitly ... 0answers 213 views ### Center of matrices over a field [duplicate] I'm trying to find the center of$\mathbb{M}_n(K)$with$K$a field. I know what the center would be if$K$was a ring, but I think this isn't the same for a field$K$. In particular I'm trying to ... 1answer 111 views ### If$L$commutes with all isomorphisms of$\mathbb{F}_2^n$, is$L=\lambda I$for some$\lambda$? [duplicate] Suppose$L$is a linear operator on a finite dimensional vector space$V$over a field of characteristic$2$. If$L\circ T=T\circ L$for all isomorphisms$T$on$V$, does this imply$L=\lambda I$for ... 3answers 99 views ### Sufficient and necessary condition for a linear map to be a scalar multiplication [duplicate] Let$V$be a finite dimensional vector space, and let$T\in\mathscr{L}(V)$(where$\mathscr{L}(V)$is the set of linear maps$V\to V$). Show that$T$is the identity multiplied by a scalar iff$TS = ...

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