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Do you see why the following two sentences use different prepositions for apparently the same functions? Can't I use the prepositions interchangeably?

1. I am sorry for shouting at you.

2. I am sorry about losing your book.
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Hi Komountain,

I use 'sorry for' when I'm admiting that I'm at fault. I use 'sorry about' when I'm expressing regret, or when I'm when I'm excusing myself.

'I'm sorry for shouting at you. There was no need for me to get so angry.'

'I'm sorry about the loss of your father.'

'I'm sorry about the mess. I'll ask someone to help me clean it.'

Here are some great explanations: [url="http://www.english-online.org.uk/adv3/sorry1a3.htm "]Advanced Course. Apologies, Lesson 3[/url]. And some exercises: [url="http://www.english-online.org.uk/adv3/sorry2a3.htm "]Advanced Course. Functions, 3[/url].
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Thank you, Matthewq.

As for my example (2), I feel 'I' is understood to be the implied agent of the action 'lose.' If I'm right, 'for' is more appropriate, because 'I' is the person at fault. If 'you' or someone else is the agent, I think 'about' is valid.

In other words,

I am sorry for losing your book. --(I lost your book.)
I am sorry about losing your book. --(You or someone else lost your book.)

Am I right?
Yes, 'for' is more appropriate in the second example. But if someone else is the agent, I'd say, 'I'm sorry about the loss of your book'. i.e. 'I regret that your book was lost.'
Thank you again for your prompt reply.

I'm still with my example (2). You mentioned 'for' is more appropriate. I agree with you, but some native speakers seem to use 'about,' even when 'I' is the agent. Is it the reflection of their intention to avoid sounding directly apologetic, or simply a misuse?

When someone else is the agent, I like your version a lot better: '...about the loss of your book.'
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As native speaker I'd use 'about' when the agent is 'I' to express regret. For instance, if I'm at a funeral, I'd tell the parents of the dead person, 'I'm sorry about your son.' Sometimes I would use it to avoid sounding directly apologetic, for example, 'I'm sorry about your car,' instead of, 'Sorry for stealing your car and smashing it into a tree.'

We use sorry for as so action and we use sorry about as so action were an object

eg I am so sorry for falling alseep at class