Let $$A = \left( {\matrix{ 1 & 2 \cr 3 & 4 \cr } } \right)$ and $B = \left( {\matrix{ a & 0 \cr 0 & b \cr } } \right),a,b \in N.$$ Then

Question

there cannot exist any $B$ such that $AB=BA$

there exist more then one but finite number of $B'$s such that $AB=BA$

there exists exactly one $B$ such that $AB=BA$

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

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