Linked Questions

Matrix multiplication is not commutative. If however $$ AB = BA $$for the matrices A and B with $$A, B \in M_{nn}(\mathbb{K})$$ Can I conclude that A has to be of the form $$A = B^{Ad} = det(B)B^{-1}$...

My friend asked the following question related to matrices: Assume A is $$ \begin{pmatrix} \frac{1}{2} & 0 & \frac{1}{2} \\ 1 & 0 & 0 \\ 0 & \frac{1}{2} & \...

Prove that for the $\mathbb R^{n\times n}$ vector space doesn't exist a basis containing $n^2$ matrices, such that AB=BA for every two matrices $A$, $B$ in the base. $(n>1)$ I think it is true ...

I'm having a hard time trying to comprehend this. How do I even start on proving the shape?

$A$ and $B$ are two matrices, when is $(A-B)(A+B)=A^2 - B^2$

Would that make things easier in any science if matrix multiplication were commutative? I mean are researchers working to find more exceptions to the general rule of matrix multiplication being not ...

Two $n\times n$ matrices $A, B$ are said to be simultaneously diagonalizable if there is a nonsingular matrix $S$ such that both $S^{-1}AS$ and $S^{-1}BS$ are diagonal matrices. a) Show that ...

Given a matrix $A$ over a field $F$, does there always exist a matrix $B$ such that $AB = BA$? (except the trivial case and the polynomial ring?)

Consider square matrices over a field $K$. I don't think additional assumptions about $K$ like algebraically closed or characteristic $0$ are pertinent, but feel free to make them for comfort. For any ...

I need to construct square matrices $A$ and $B$ such that $AB=0$ but $BA \neq 0$. I know matrix multiplication is not commutative, but I don't know how to construct such matrices. Thanks in advance. ...

Let $A = \begin{pmatrix} 1 & 1& 1\\ 1 & 2 &3 \\ 1 &4 & 5 \end{pmatrix}$ and $D = \begin{pmatrix} 2 & 0& 0\\ 0 & 3 &0 \\ 0 &0 & 5 \end{pmatrix}$....

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? ...

I have a question asking me why matrix multiplication isn't commutative. I'm not exactly sure what's the best way to explain this without simply saying "it's obvious". $AB \not= BA$ because ...

If there are two matrices A and B that are both nxn matrices, will AB = BA always? Is there a way to have those two matrices so that AB = 0 but BA ≠ 0?

This is a very old and popular question: the following link is the similar question When is matrix multiplication commutative? And there is a very famous theorem: If $A,B$ are simultaneously ...