If R be the ring of 2 xx 2 complex matrices. When is the left annihilator just equal to {0}?

Annie French

Annie French

Answered question

2022-11-17

If we let R be the ring of 2 × 2 complex matrices. When is the left annihilator just equal to {0}? I see that if A is invertible Ann R ( A ) is trivial since if M Ann R ( A ) then M A = 0 so we can just multiply on the right by A 1 and so M = 0

Answer & Explanation

kuthiwenihca

kuthiwenihca

Beginner2022-11-18Added 23 answers

If A is not invertible, then consider a matrix V associated to some non-zero linear map that is 0 on col A. Then, V A = 0 and V 0
Anton Huynh

Anton Huynh

Beginner2022-11-19Added 5 answers

Every non-invertible matrix has a matrix which annihilates it. In general it can be constructed with the pseudo-inverse. In the 2 × 2 case the non-invertible:
[ a b c d ]
is annihilated by its adjoint:
[ d b c a ]

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