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Maths-

General

Easy

Question

Let A = $open parentheses table row 1 2 row 3 4 end table close parentheses$ and B = $open parentheses table row a 0 row 0 b end table close parentheses$ , a, b $element of$ N. Then

there exist more than one but finite number of B's such that AB = BA
there exist exactly one B such that AB = BA
there exist infinitely many B's such that AB = BA
there cannot exist any B such that AB = BA

hint Hint:

$G i v e n space A equals open square brackets table row 1 2 row 3 4 end table close square brackets space B equals open square brackets table row a 0 row 0 b end table close square brackets F i n d i n g space A B A B equals open square brackets table row cell a plus 2 cross times 0 end cell cell 1 cross times 0 plus 2 cross times b end cell row cell 3 x a plus 4 x 0 end cell cell 3 x 0 plus 4 x b end cell end table close square brackets A B equals open square brackets table row a cell 2 b end cell row cell 3 a end cell cell 4 b end cell end table close square brackets F i n d i n g space B A B A equals open square brackets table row cell a plus 0 x 3 end cell cell 2 a plus 0 x 4 end cell row cell 0 x 1 plus 3 b end cell cell 0 x 2 plus 4 b end cell end table close square brackets B A equals open square brackets table row a cell 2 a end cell row cell 3 b end cell cell 4 b end cell end table close square brackets I f space A B equals B A open square brackets table row a cell 2 b end cell row cell 3 a end cell cell 4 b end cell end table close square brackets equals open square brackets table row a cell 2 a end cell row cell 3 b end cell cell 4 b end cell end table close square brackets C o m p a r i n g space e a c h space e l e m e n t space w e space g e t a equals a comma 2 b equals 2 a rightwards double arrow a equals b therefore T h e r e space a r e space i n f i n i t e l y space m a n y space b apostrophe s space f o r space w h i c h space A B equals B A$

The correct answer is: there exist infinitely many B's such that AB = BA

Given the matrices A and B we need to find the correct option.

Therefore, there are infinitely many b's for which AB=BA

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