Mjólkin er skemmd, svo ég kaupi nýja.

In Icelandic, the definite article is usually attached to the end of the noun instead of being a separate word.

• mjólk = milk
• mjólkin = the milk

So the ending -in here is the definite article.

Mjólkin is in the nominative singular.

That is because it is the subject of the sentence: it is the thing that is spoiled.

So:

• mjólkin = subject
• therefore it appears in the nominative

Because skemmd has to agree with mjólkin.

Mjólk is:

• feminine
• singular
• here, nominative

So the adjective/past participle must also be feminine singular nominative:

• masculine: skemmdur
• feminine: skemmd
• neuter: skemmt

Since mjólkin is feminine, skemmd is the correct form.

It is historically a past participle of að skemma (to damage / spoil), but in sentences like this it behaves very much like an adjective.

That is very common in Icelandic. English does something similar in sentences like:

• The milk is spoiled
• The door is closed

So in practice, a learner can treat skemmd here as an adjective-like form that agrees with the noun.

No. Skemmdur / skemmd / skemmt is a broader word meaning something like damaged, ruined, spoiled, broken, depending on the noun.

So with food, it often means spoiled.
With an object, it might mean damaged or broken.

Context tells you which English word sounds best.

Here svo means so, therefore, or as a result.

It connects the two ideas:

• The milk is spoiled
• so I buy a new one / new milk

So it is showing cause and result.

Because kaupi is the conjugated verb, while kaupa is the infinitive.

The verb is að kaupa = to buy.

Present tense:

• ég kaupi = I buy
• þú kaupir = you buy
• hann/hún/það kaupir = he/she/it buys

So ég kaupi is simply the correct 1st person singular present tense form.

It can mean either, depending on context.

Icelandic often uses the present tense where English might use will for a near-future decision or plan.

So in this sentence, ég kaupi nýja can naturally mean:

• I buy a new one
• or I’ll buy a new one

If the situation is clearly about what you are about to do, the future sense is very natural.

Because nýja agrees with an understood noun, and that noun is mjólk.

After kaupa, the object is in the accusative, so the hidden noun is understood as:

• mjólk = feminine singular
• object of kaupa = accusative

The adjective therefore has to be feminine singular accusative, which is nýja.

If you said the noun out loud, it would be:

• ég kaupi nýja mjólk

Icelandic often leaves the noun out when it is obvious from context.

So nýja here really means something like:

• new milk
• a new one, with mjólk understood

English often needs one or a repeated noun, but Icelandic can just use the adjective by itself when the meaning is clear.

Because this is an indefinite idea: new milk / a new one, not the new one.

Icelandic normally uses:

• strong adjective forms for indefinite noun phrases
• weak adjective forms for definite ones

So nýja is the strong feminine accusative singular form, which fits this sentence.

In this sentence, svo is functioning as a conjunction meaning so, linking two clauses. After that, the second clause keeps normal clause order:

• ég kaupi nýja

If svo were being used more like then, you could also get a structure like svo kaupi ég... with a slightly different feel.

So for this sentence, svo ég kaupi nýja is a natural way to say so I buy / so I’ll buy a new one.