Learn Hardcore Icelandic: Tómaturinn er rauður.

tómatur → tómaturinn = the tomato For masculine nouns ending in -ur, the definite nominative singular is typically -urinn: tómatur + inn → tómaturinn.
When an adjective comes before a definite noun, the adjective takes the weak ending and the article still sits on the noun: rauði tómaturinn = the red tomato. (A separate article hinn exists, e.g., hinn rauði tómatur, but it’s formal/literary.)

Why is it rauður and not some other form?

Adjectives agree with the noun’s gender, number, and case. Tómaturinn is masculine, singular, nominative, so the predicate adjective appears as masculine singular nominative, strong: rauður.
In predicative position (after er “is”), adjectives normally use the strong declension.

Could I say Tómaturinn er rauði?

No in standard usage. Rauði is the weak form, used attributively with a definite noun (e.g., rauði tómaturinn “the red tomato”). In predicative position after er, use the strong form: rauður.
So: Tómaturinn er rauður, but Rauði tómaturinn er góður (“The red tomato is good”).

What are the forms of “red” across genders and number?

Masculine singular: rauður (as in the sentence)
Feminine singular: rauð (e.g., Bókin er rauð “The book is red”)
Neuter singular: rautt (e.g., Húsið er rautt “The house is red”)
Masculine plural: rauðir (e.g., Tómatar eru rauðir “Tomatoes are red”)
Feminine plural: rauðar
Neuter plural: rauð

How do I say “Tomatoes are red” and “The tomatoes are red”?

“Tomatoes are red.” → Tómatar eru rauðir.
“The tomatoes are red.” → Tómatarnir eru rauðir.

How would I express the indefinite idea “A tomato is red”?

Icelandic has no indefinite article. You simply use the bare noun:

“A tomato is red.” → Tómatur er rauður.
Using the definite form Tómaturinn usually points to a specific tomato, not a generic statement. For a generic claim, plural is common: Tómatar eru rauðir.

Which case is tómaturinn in here, and what about the adjective?

tómaturinn is in the nominative case (it’s the subject).
The predicate adjective rauður also appears in nominative and agrees with the subject in gender and number.
If tomato were an object, you’d see accusative:
“I’m eating the tomato.” → Ég borða tómatinn. (accusative singular definite)

How do I turn the sentence into a yes–no question?

Invert subject and verb:

Er tómaturinn rauður? = “Is the tomato red?”
Answer with Já (yes) / Nei (no):
Já, tómaturinn er rauður. / Nei, tómaturinn er ekki rauður.

How do I negate it?

Place ekki (not) after the verb:

Tómaturinn er ekki rauður. = “The tomato is not red.”

Can I front the adjective for emphasis?

Yes, for emphasis or contrast you can front the predicate adjective:

Rauður er tómaturinn. ≈ “Red is the tomato.” (emphatic/stylistic)
The neutral order is Tómaturinn er rauður.

How is it pronounced? Any tips for ó, au, and ð?

Tómaturinn: stress first syllable; ó is like a long “oh”; final -ur has a short, rounded vowel (like the u in English “put”) plus a trilled r.
er: like “air” but short.
rauður: au is a diphthong roughly like English “oi” in “boil” but with rounded lips; ð is the voiced “th” in “this”.
Very rough guide: TOH-ma-tur-in air ROY-thur (with a trilled r).

What’s the present-tense pattern of að vera (to be) relevant to this?

ég er (I am)
þú ert (you are, singular)
hann/hún/það er (he/she/it is)
við erum (we are)
þið eruð (you are, plural)
þeir/þær/þau eru (they are)
Here we use er because the subject is third person singular.

Any capitalization or spelling gotchas?

Nouns are not capitalized in Icelandic (only sentence-initial words and proper names). Tómaturinn is capitalized here solely because it starts the sentence.
Note the letters: ó (not plain o), ð (voiced “th”), and the diphthong au in rauður.