После искреннего извинения мой брат быстро простил сестру.

Because после always takes the genitive case in Russian.

• Dictionary form: искреннее извинение = a sincere apology
• After после: после искреннего извинения = after a sincere apology

Here is the case change:

• извинение → извинения
• искреннее → искреннего

So the whole phrase means after a sincere apology, with both the noun and adjective in the genitive singular.

Because it agrees with извинения, which is genitive singular.

The noun извинение is neuter, and in the genitive singular, neuter adjectives take the same ending as masculine adjectives:

• nominative: искреннее извинение
• genitive: искреннего извинения

This is a very common pattern in Russian:

• новое письмо → без нового письма
• искреннее извинение → после искреннего извинения

So -его here is just the normal genitive singular adjective ending.

Because they have different roles in the sentence.

• мой брат is the subject → nominative case
• сестру is the direct object → accusative case

So:

• брат stays nominative
• сестра changes to сестру in the accusative singular

This happens with many feminine nouns ending in -а:

• мама → маму
• девочка → девочку
• сестра → сестру

So мой брат быстро простил сестру means my brother quickly forgave his sister / the sister.

Because простить usually takes a direct object in the accusative when it means to forgive someone.

So:

• простить кого? → сестру
• not сестре

Examples:

• Я простил друга. = I forgave my friend.
• Она простила мать. = She forgave her mother.

Russian uses the accusative here where English also uses a direct object: forgive someone.

This is a very important distinction.

• простить = to forgive
• извиниться = to apologize
• извинить can mean to excuse / forgive, but in modern everyday Russian, простить is the normal verb for forgiving someone

In your sentence:

• извинение = apology
• простил = forgave

So the structure is:

After a sincere apology, my brother quickly forgave his sister.

A learner should not mix these up:

• Она извинилась. = She apologized.
• Он простил её. = He forgave her.

Because простил is perfective, and it presents the forgiveness as a completed event.

• простил = forgave / did forgive
• прощал = was forgiving, used to forgive, or describes a repeated/process meaning

In this sentence, the action is a single completed result:

• apology happened
• then forgiveness happened

So простил is the natural choice.

If you used прощал, it would sound like an ongoing or habitual action, which does not fit this context well.

Быстро means quickly, and here it modifies the verb простил.

So it tells us how the brother forgave his sister:

• мой брат простил сестру = my brother forgave his sister
• мой брат быстро простил сестру = my brother quickly forgave his sister

In other words, he did not take long to forgive her.

No. Russian word order is relatively flexible because case endings show grammatical roles.

The neutral order here is:

После искреннего извинения мой брат быстро простил сестру.

But other orders are possible, for example:

• Мой брат быстро простил сестру после искреннего извинения.
• Сестру мой брат быстро простил после искреннего извинения.

These alternatives may sound more marked or emphasize different parts, but the basic meaning remains similar.

The original sentence sounds natural and fairly neutral, with the time/background phrase после искреннего извинения placed first.

The Russian sentence does not explicitly say.

It simply says after a sincere apology. From context, most people will probably assume it was the sister who apologized, since she is the one being forgiven. But grammatically, the sentence itself does not force that interpretation.

If Russian wanted to make it explicit, it could say something like:

• После её искреннего извинения... = After her sincere apology...
• После искреннего извинения сестры... = After the sister’s sincere apology...

So in your sentence, the apologizer is understood from context, not from grammar alone.

Yes, depending on context.

Извинение can mean:

• apology
• sometimes excuse or request for forgiveness

In this sentence, because of простил сестру, the meaning is clearly apology.

So:

• искреннее извинение here = a sincere apology

This is a good example of how context helps choose the right English translation.

Russian has no articles like a or the.

So сестру by itself can mean:

• a sister
• the sister
• sometimes his sister / her sister, depending on context

In this sentence, English would most naturally say his sister, because мой брат is the subject and the family relationship is obvious from context:

After a sincere apology, my brother quickly forgave his sister.

Russian often leaves this kind of relationship unstated when it is clear enough.

Yes, often you can.

Russian often omits possessive words like мой, его, её when the meaning is clear from context. So:

• Мой брат быстро простил сестру.
• Брат быстро простил сестру.

Both are possible.

Using мой makes it explicit that you are talking about my brother. Without it, the sentence may sound more general or depend more heavily on context.

So мой is not grammatically required, but it is perfectly natural if you want to be clear.

A helpful breakdown is:

• После искреннего извинения = background/time phrase: after a sincere apology
• мой брат = subject: my brother
• быстро = adverb: quickly
• простил = verb: forgave
• сестру = direct object: (his/the) sister

So the sentence structure is basically:

After + [genitive phrase], subject + adverb + verb + object

This is a very common Russian pattern.