"However" versus "No matter"

[Tom]

Hi

Could you please tell me if the given pairs of sentences mean the same thing?

1- However short he may be, he looks handsome.
2- No matter how short he is, he looks handsome.

1- However fast you may run, you will not reach there on time.
2- No matter how fast you run, you will not reach there on time.

1-However roughly she may speak, I cannot be rude to her.
2- No matter how roughly she may speak, I cannot be rude to her.

Tom

[Mister Micawber]

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They seem essentially synonymous to me.
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Native English teacher at Mister Micawber's