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

1. there cannot exist any B such that AB = BA
2. there exist more than one but finite number of B's such that AB = BA
3. there exists exactly one B such that AB = BA
4. there exist infinitely many B's such that AB = BA

Correct answer: there exist infinitely many B's such that AB = BA

Solution

The matrices A and B commute if their product is the same regardless of the order of multiplication. Since B is a diagonal matrix, it can take on infinitely many combinations of the natural numbers for a and b, allowing for an infinite number of matrices B that satisfy the commutation relation with A.

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