الغرفة المزدوجة كبيرة.

In Arabic, descriptive adjectives normally come after the noun they describe.

So:

• الغرفة = the room
• المزدوجة = the double

Together, الغرفة المزدوجة literally follows the pattern the room the-double, which in natural English becomes the double room.

Because Arabic adjectives must agree with the noun they describe in definiteness.

• الغرفة has الـ, so it is definite: the room
• therefore the adjective must also be definite: المزدوجة

This is the normal rule for an adjective directly modifying a noun.

Compare:

• غرفة مزدوجة = a double room
• الغرفة المزدوجة = the double room

Because كبيرة is not part of the phrase الغرفة المزدوجة. It is the predicate of the sentence, meaning big in The double room is big.

In Arabic, the predicate in this kind of sentence is usually indefinite, so it normally does not take الـ.

So:

• الغرفة المزدوجة = the double room
• كبيرة = is big

If you said الكبيرة here, it would usually sound like you were making it into another definite description rather than the sentence predicate.

Arabic usually leaves out the verb to be in the present tense.

So:

• الغرفة المزدوجة كبيرة literally looks like the double room big
• but it means The double room is big

This is completely normal in Arabic.

Because they are both feminine forms.

The noun غرفة is feminine, so adjectives connected to it must also be feminine:

• مزدوج = masculine

• مزدوجة = feminine

• كبير = masculine
• كبيرة = feminine

So Arabic is showing agreement with الغرفة.

Usually, yes. The ending ـة is called taa marbuuTa, and it is a very common sign of a feminine noun.

So for a learner, seeing غرفة strongly suggests that the noun is feminine, and in this case that is correct.

Not every feminine noun ends this way, and not every word ending this way behaves exactly the same in every context, but it is a very useful clue.

It is a nominal sentence in Arabic.

You can divide it like this:

• الغرفة المزدوجة = the subject/topic
• كبيرة = the predicate/comment

So the sentence pattern is:

[definite noun phrase] + [predicate adjective]

That gives the meaning The double room is big.

With full vowel endings, it would be:

الغرفةُ المزدوجةُ كبيرةٌ

Why?

• الغرفةُ: subject, so it takes ـُ
• المزدوجةُ: adjective matching the noun
• كبيرةٌ: indefinite predicate, so it takes tanween

In normal everyday Arabic writing, these short endings are usually omitted.

In these words, the ل in الـ is pronounced clearly, because the next letters are moon letters, not sun letters.

So you pronounce:

• الغرفة as al-ghurfa
• المزدوجة as al-muzdawija

There is no merging of the ل here.

Here it means double or for two, as in hotel vocabulary.

So:

• غرفة مزدوجة = a double room
• الغرفة المزدوجة = the double room

It is the feminine form because it describes غرفة, which is feminine.

That final ـة is taa marbuuTa. In pause, it is usually pronounced like -a.

So you normally hear:

• الغرفة ≈ al-ghurfa
• المزدوجة ≈ al-muzdawija
• كبيرة ≈ kabīra

But when full grammatical endings are pronounced in careful reading, the t sound can appear before the ending:

• الغرفةُ = al-ghurfatu
• المزدوجةُ = al-muzdawijatu
• كبيرةٌ = kabīratun

So both ideas are related to the same spelling.