Little task about matrix I don't quite understand

Question

Here is another task from exam but this time I don't understand how to solve it. So it says:

Show that: If $A \in \mathbb{R}^{2\times 2}$ is a matrix of the from $A=\begin{pmatrix} \lambda & 0\\ 0 & \lambda \end{pmatrix}$ where $\lambda \in \mathbb{R}$, then for all $B \in \mathbb{R}^{2\times 2}$ we have that the identity is $A \cdot B = B \cdot A$

I think similar task will be asked in our exam but I need to understand this.

So $\mathbb{R}^{2 \times 2}$ means we have matrix with 2 rows and 2 columns? And the numbers inside this matrix are real? Is that what's meant by that?

So they basically ask if we multiply $A$ with any real matrix $B$ (2 rows 2 columns), then we have that $AB = BA$? Or I understood wrong?

But then I wouldn't know how to reason it, but the zeroes provide us same result I think.

Answer 1

Always remember the only matrices that commutes with all other matrices are the multiples of the identity. Here $A $ satisfies this condition, so for $B \in \mathbb R^{2\times 2}$ $A \cdot B = B \cdot A $.

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Also, in general, matrix multiplication is commutative when the two are diagonal matrices and are of the same dimension. Hope it helps.