## The vector space of commuting matrices

### Task number: 2511

Let $$\mathbf D$$ be a square matrix over a field $$\mathbb K$$. Show that all the matrices which commute in matrix product with matrix $$\mathbf D$$ form a vector space.

• #### Hint

Show that it is a subspace of the vector space of all square matrices over the field $$\mathbb K$$.

• #### Resolution

All zero matrix commutes trivially with any matrix. Lets denote the set of all $$\mathbf D$$-commutable matrices as $$C$$.

Let $$\mathbf A, \mathbf B \in C$$, then $$(\mathbf A + \mathbf B)\cdot\mathbf D = \mathbf A\mathbf D + \mathbf B\mathbf D = \mathbf D\mathbf A + \mathbf D\mathbf B = \mathbf D\cdot(\mathbf A + \mathbf B)$$

Now let $$\mathbf A \in C$$ and $$a\in \mathbb K$$, then $$(a\mathbf A)\cdot\mathbf D = a(\mathbf A\cdot\mathbf D) = a(\mathbf D\cdot\mathbf A) = \mathbf D\cdot(a\mathbf A)$$