Question

Let $straight A space equals space open parentheses table row 1 2 row 3 4 end table close parentheses space and space straight B space equals open parentheses table row straight a 0 row 0 straight b end table close parentheses$ a , b ∈ N. Then

there cannot exist any B such that AB = BA

there exist more than 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

Solution

there exist infinitely many B’s such that AB = BA

straight A space equals open square brackets table row 1 2 row 3 4 end table close square brackets space space space space straight B space equals open square brackets table row straight a 0 row 0 straight b end table close square brackets AB space equals open square brackets table row straight a cell 2 straight b end cell row cell 3 straight a end cell cell 4 straight b end cell end table close square brackets BA space equals space open square brackets table row straight a 0 row 0 straight b end table close square brackets open square brackets table row 1 2 row 3 4 end table close square brackets space equals space open square brackets table row straight a cell 2 straight a end cell row cell 3 straight b end cell cell 4 straight b end cell end table close square brackets

AB = BA only when a = b

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