Linked Questions

3
votes
4answers
2k views

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? ...
0
votes
1answer
5k views

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) $
3
votes
2answers
4k views

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 ...
1
vote
2answers
2k views

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 $A = \lambda E_2$. ...
3
votes
2answers
2k views

Center of general linear group [duplicate]

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\...
0
votes
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 ...
0
votes
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? ...
3
votes
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 $T= \lambda I$, ...
1
vote
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 ...
2
votes
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 $fg = gf$ ...
2
votes
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 $A=\lambda I_n$
3
votes
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 ...
0
votes
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 ...
1
vote
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 ...
2
votes
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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