Let A=1234, B=a00b, a,b∈N then

AB=BA 1234 a00b=a00b 1234 a2b3a4b=a2a3b4b ∴ 2b=2a & 3a=3b a=b & a=b ∴ There exist infinitely many B’s such that AB=BA

Let A=1234, B=a00b, a,b∈N then

AB=BA 1234 a00b=a00b 1234 a2b3a4b=a2a3b4b ∴ 2b=2a & 3a=3b a=b & a=b ∴ There exist infinitely many B’s such that AB=BA

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Let $A = (\begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix}), B = (\begin{matrix} a & 0 \\ 0 & b \end{matrix}), a, b \in N$ then

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There cannot exist any B such that AB=BA

There exists exactly one B such that AB=BA

There exist more than one but finite number of B's such that AB = BA

There exist infinitely many B's such that AB=BA

answer is D.

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Detailed Solution

$AB = BA [\begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix}] [\begin{matrix} a & 0 \\ 0 & b \end{matrix}] = [\begin{matrix} a & 0 \\ 0 & b \end{matrix}] [\begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix}] [\begin{matrix} a & 2 b \\ 3 a & 4 b \end{matrix}] = [\begin{matrix} a & 2 a \\ 3 b & 4 b \end{matrix}] ∴ 2 b = 2 a & 3 a = 3 b a = b & a = b$