Let A=[1 2 3 4] and B=[a 0 0 b] a, b ∈ N. Then,

Question

Let A=[1 2 3 4] and B=[a 0 0 b] a, b ∈ N. Then, (A) there exists more than one but finite number of B prime s such that A B=B A (B) there exists exactly one B such that A B=B A (C) there exists infinitely many B prime s such that A B=B A (D) there cannot exist any B such that A B=B A

Tardigrade Admin · Accepted Answer

Given that, A =[1 2 3 4] and B B=[a 0 0 b] Now, A B=[1 2 3 4][a 0 0 b]=[a 2 b 3 a 4 b] and B A=[a 0 0 b][1 2 3 4]=[a 2 a 3 b 4 b] If A B=B A, then a=b Hence, A B=B A is possible for infinitely many values of B.

Q. Let $A = [1324]$ and $B = [a 0 0 b] a, b \in N$ . Then,

there exists more than one but finite number of $B^{'}$ s such that $A B = B A$

there exists exactly one $B$ such that $A B = B A$

there exists infinitely many $B^{'}$ s such that $A B = B A$

there cannot exist any $B$ such that $A B = B A$

Given that,
$A = [1324]$
and $BB = [a 0 0 b]$
Now, $A B = [1324] [a 0 0 b] = [a 3 a 2 b 4 b]$
and $B A = [a 0 0 b] [1324] = [a 3 b 2 a 4 b]$
If $A B = B A$ , then $a = b$
Hence, $A B = B A$ is possible for infinitely many values of $B$ .

Q. Let A=[13​24​] and B=[a0​0b​]a,b∈N. Then,

there exists more than one but finite number of B′ s such that AB=BA

there exists exactly one B such that AB=BA

there exists infinitely many B′ s such that AB=BA

there cannot exist any B such that AB=BA