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(D) Given $A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}$ and $B = \begin{bmatrix} a & 0 \ 0 & b \end{bmatrix}$.Calculate $AB$:$AB = \begin{bmatri

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there exist infinitely many $B$'s such that $AB = BA$

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(D) Given $A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}$ and $B = \begin{bmatrix} a & 0 \ 0 & b \end{bmatrix}$.Calculate $AB$:$AB = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} \begin{bmatrix} a & 0 \ 0 & b \end{bmatrix} = \begin{bmatrix} a & 2b \ 3a & 4b \end{bmatrix}$.Calculate $BA$:$BA = \begin{bmatrix} a & 0 \ 0 & b \end{bmatrix} \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} = \begin{bmatrix} a & 2a \ 3b & 4b \end{bmatrix}$.For $AB = BA$,we must have:$\begin{bmatrix} a & 2b \ 3a & 4b \end{bmatrix} = \begin{bmatrix} a & 2a \ 3b & 4b \end{bmatrix}$.Comparing the elements,we get $2b = 2a$ and $3a = 3b$,which both imply $a = b$.Since $a, b \in \mathbb{N}$,there are infinitely many pairs $(a, b)$ such that $a = b$ (e.g.,$(1, 1), (2, 2), (3, 3), \dots$).Therefore,there exist infinitely many matrices $B$ such that $AB = BA$.

Let $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ and $B = \begin{bmatrix} a & 0 \\ 0 & b \end{bmatrix}$,where $a, b \in \mathbb{N}$. Then:

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