Az intézet nagy.

Hungarian has two forms of the definite article:

• a before a consonant sound
• az before a vowel sound

Because intézet begins with a vowel, Hungarian uses az intézet.

Compare:

• a ház = the house
• az intézet = the institute

This is similar to English a/an in function, although Hungarian uses it only for the definite article.

In Hungarian, the verb to be is usually left out in the present tense, third person when the sentence is simple and descriptive.

So:

• Az intézet nagy. = The institute is big.

But in other persons, Hungarian does use the verb:

• Nagy vagyok. = I am big.
• Nagy vagy. = You are big.
• Nagy. = He/She/It is big.

It also appears in the past or future:

• Az intézet nagy volt. = The institute was big.
• Az intézet nagy lesz. = The institute will be big.

So the missing is is completely normal here.

When an adjective directly modifies a noun, it usually comes before the noun:

• nagy intézet = a big institute

But in your sentence, nagy is not directly modifying the noun inside a noun phrase. It is the predicate of the sentence:

• Az intézet nagy. = The institute is big.

So this is not big institute, but rather the institute ... big.

That is why nagy appears after intézet here.

Yes. This is the most neutral, straightforward word order for this sentence.

A very common basic pattern in Hungarian is:

• Subject + predicate

So here:

• Az intézet = subject
• nagy = predicate adjective

Hungarian word order is flexible, but changes in word order often affect focus or emphasis. For a beginner, Az intézet nagy. is the standard version to learn first.

A few useful pronunciation points:

• Hungarian stress is usually on the first syllable of each word.
• sz is pronounced like English s
• gy is a soft sound, somewhat like the d in British duke or a softened dy
• á, é are long vowels

A rough guide:

• Az → like oz with a short vowel
• intézet → EEN-tay-zet
• nagy → roughly nody or nawdy, but with a special soft gy

So the whole sentence is approximately:

• az EEN-tay-zet nodj

That is only an approximation, but it helps at the start.

No. Hungarian does not have grammatical gender like many European languages do.

So nouns are not masculine, feminine, or neuter. That means:

• no gendered articles
• no adjective agreement by gender
• no need to memorize noun gender

This is one reason Hungarian grammar can feel simpler than languages like German or French in this area.

These are basically the plain base forms:

• intézet = the noun in its basic singular form
• nagy = the adjective in its basic form

The only extra element is the definite article az.

There is no case ending here on intézet, because it is the subject of the sentence, and Hungarian subjects normally appear in the basic form.

The plural would be:

• Az intézetek nagyok. = The institutes are big.

Here two things change:

• intézet → intézetek = plural noun
• nagy → nagyok = plural predicate adjective

This is worth noticing: in Hungarian, when an adjective is used as the predicate, it often shows plurality too.

So:

• Az intézet nagy. = The institute is big.
• Az intézetek nagyok. = The institutes are big.

Yes, but that does not mean the same thing as the full sentence.

• a nagy intézet = the big institute
  This is just a noun phrase, not a complete sentence.

• Az intézet nagy. = The institute is big.
  This is a complete sentence.

So the difference is:

• nagy intézet = adjective directly before noun
• intézet nagy = noun plus predicate adjective in a sentence

That distinction is very important in Hungarian.