If A = begin{bmatrix} 3 & 4 5 & 6 end{bmatri | vedclass

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(D) Given $A = \begin{bmatrix} 3 & 4 \ 5 & 6 \end{bmatrix}$ and $B = \begin{bmatrix} x & 0 \ 0 & y \end{bmatrix}$.Calculating $AB$:$AB = \begin{bmat

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There exist infinite number of matrices $B$ such that $AB = BA$

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(D) Given $A = \begin{bmatrix} 3 & 4 \ 5 & 6 \end{bmatrix}$ and $B = \begin{bmatrix} x & 0 \ 0 & y \end{bmatrix}$.Calculating $AB$:$AB = \begin{bmatrix} 3 & 4 \ 5 & 6 \end{bmatrix} \begin{bmatrix} x & 0 \ 0 & y \end{bmatrix} = \begin{bmatrix} 3x & 4y \ 5x & 6y \end{bmatrix}$.Calculating $BA$:$BA = \begin{bmatrix} x & 0 \ 0 & y \end{bmatrix} \begin{bmatrix} 3 & 4 \ 5 & 6 \end{bmatrix} = \begin{bmatrix} 3x & 4x \ 5y & 6y \end{bmatrix}$.For $AB = BA$,we must have:$3x = 3x$ (always true)$4y = 4x \Rightarrow x = y$$5x = 5y \Rightarrow x = y$$6y = 6y$ (always true)Thus,$AB = BA$ if and only if $x = y$.Since $x, y \in \mathbb{N}$,we can choose any natural number $n$ such that $x = y = n$. Since there are infinitely many natural numbers,there exist infinitely many such matrices $B$.

If $A = \begin{bmatrix} 3 & 4 \\ 5 & 6 \end{bmatrix}$ and $B = \begin{bmatrix} x & 0 \\ 0 & y \end{bmatrix}$,where $x, y \in \mathbb{N}$,then:

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