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Let A = [1 2; 3 4] and B = [a 0; 0 b], where a, b are natural numbers. Then

1. no such matrix B can satisfy AB = BA
2. there are several, but only finitely many, matrices B for which AB = BA
3. there is exactly one matrix B for which AB = BA
4. there are infinitely many matrices B for which AB = BA

Correct answer: there are infinitely many matrices B for which AB = BA

Solution

With B = diag(a,b), AB = BA requires 2b=2a and 3a=3b, hence a=b. Then B = aI commutes with A for every natural number a, so there are infinitely many such matrices B.

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