Banken undersøger sagen.

Because -en is the definite article attached to the noun.

• bank = a bank
• banken = the bank

Danish often puts the at the end of the noun instead of using a separate word like English does.

For the same reason: -en makes the noun definite.

• sag = a case / a matter
• sagen = the case / the matter

So both nouns in the sentence are definite:

• banken = the bank
• sagen = the case

Undersøger is the present tense of the verb at undersøge.

• at undersøge = to investigate / to examine
• undersøger = investigates / is investigating

In Danish, the present tense is often formed with -r.

Examples:

• jeg undersøger = I investigate / am investigating
• banken undersøger = the bank investigates / is investigating

It can mean both, depending on context.

Danish present tense often covers both:

• a general present: investigates
• an ongoing action: is investigating

So Banken undersøger sagen could mean either:

• The bank investigates the case
• The bank is investigating the case

Usually the wider context tells you which is meant.

This is the normal Danish subject–verb–object order in a main clause:

• Banken = subject
• undersøger = verb
• sagen = object

So it works just like basic English word order:

• The bank
  • investigates
    • the case

No. Danish verbs do not change according to the subject in the present tense.

So you say:

• jeg undersøger
• du undersøger
• han undersøger
• banken undersøger

The verb stays undersøger in all these cases. That is much simpler than English.

Both bank and sag are common gender nouns in Danish, and that is why their definite singular form is -en:

• en bank → banken
• en sag → sagen

This matters because Danish nouns take different articles and endings depending on gender.

Very roughly:

• common gender: en, definite -en
• neuter: et, definite -et

Usually not in the same basic sense.

• banken simply means the bank
• den bank usually means that bank or is used in special structures

So if you just mean the bank, banken is the natural form.

Not always. Sag is a flexible word and can mean things like:

• case
• matter
• issue
• affair

In Banken undersøger sagen, the case or the matter are both natural translations. The best choice depends on context.

The ø sound does not exist exactly in standard English, so learners often need practice with it.

A rough guide:

• ø is a rounded front vowel
• your tongue is somewhat like for e in bed, but with rounded lips

So undersøger is not pronounced like English under-so-ger.

A rough learner-friendly approximation might be:

• UN-der-suh-yer or UN-er-suh-yer

But that is only approximate. The real Danish pronunciation is smoother, and some sounds are softer than English spelling suggests.

Because Danish commonly expresses the by adding it to the noun itself.

So instead of:

• the bank
• the case

Danish says:

• banken
• sagen

This is one of the most important differences from English. In simple sentences like this, the definite ending usually does the job on its own.

It can mean either, depending on context, but in this sentence it is most naturally understood as the bank as an institution.

That is because the verb undersøger suggests an organization carrying out an investigation. A building cannot literally investigate something, but a bank as a company or institution can.

So here banken is best understood as the bank in the institutional sense.

You would use the indefinite noun forms:

• En bank undersøger en sag.

That means:

• A bank is investigating a case or
• A bank investigates a case

Compare:

• Banken undersøger sagen = The bank is investigating the case
• En bank undersøger en sag = A bank is investigating a case