Mater infantem nudum in balneo lavat.

Because infantem is the accusative singular form, and it is the direct object of lavat.

• infans = the infant/baby as a subject
• infantem = the infant/baby as the object

So in this sentence:

• Mater = the mother → subject
• infantem = the baby → object

Latin shows the job of a noun by its ending, not mainly by word order.

Because nudum has to agree with infantem.

The adjective nudus, -a, -um means naked. Here it describes infantem, which is:

• accusative
• singular
• masculine in this sentence

So the adjective must also be:

• accusative
• singular
• masculine

That gives nudum.

If the baby were grammatically feminine, you would expect nudam instead.

Grammatically, yes: nudum is masculine accusative singular, so the infant is being treated as masculine in this sentence.

The noun infans can refer to either a male or female child, but the adjective tells you how the noun is being treated here. Since the adjective is masculine, the baby is understood as male, or at least grammatically masculine.

Because in with the ablative usually means in or inside, showing location.

• in balneo = in the bath / in the bathroom / in the bathing place

Latin uses:

• in + ablative for where
• in + accusative for into where, showing motion toward something

So:

• in balneo = in the bath
• in balneum = into the bath

Here the mother is washing the baby in the bath, not moving the baby into it.

Balneo is ablative singular.

Its dictionary form is usually given as balneum, balnei (neuter), meaning bath or bathroom.

So the forms are something like:

• balneum = nominative/accusative singular
• balnei = genitive singular
• balneo = dative/ablative singular

In this sentence, balneo is ablative because it follows in to show location.

In this sentence, mater is the subject because it is in the nominative singular form.

The important thing in Latin is usually the case ending, not the position in the sentence. Here:

• mater = nominative singular → subject
• infantem = accusative singular → direct object

So even if the words were rearranged, the endings would still show who is doing the action and who is receiving it.

Yes, but it is also very normal.

Mater infantem nudum in balneo lavat is a natural Latin word order:

• subject: Mater
• object + adjective: infantem nudum
• prepositional phrase: in balneo
• verb at the end: lavat

Latin often places the verb last, though it does not have to.

You could also see other orders, such as:

• Mater in balneo infantem nudum lavat
• Infantem nudum mater in balneo lavat

These would still mean basically the same thing, though the emphasis may shift slightly.

Because Latin has no articles.

So mater can mean:

• mother
• the mother
• sometimes even a mother

And infantem can mean:

• an infant
• the infant
• the baby

English needs articles, but Latin usually does not use separate words for them. You understand from context whether it is a or the.

It can mean either, depending on context.

Lavat is present tense, third person singular of lavare.

So it can be translated as:

• washes
• is washing

Latin does not make the same regular distinction that English does between simple present and present progressive. The context decides which English version sounds best.

Lavat is:

• present tense
• active voice
• indicative mood
• third person singular

It comes from lavare, meaning to wash.

So lavat means:

• he washes
• she washes
• it washes

Here, since mater is the subject, it means she washes.

Because the verb ending already tells you the person and number.

The ending -t in lavat means:

• third person singular

So Latin does not need to add a separate pronoun like she unless it wants extra emphasis.

That is why mater lavat already clearly means the mother washes or she washes.

It is an ordinary adjective, but in this sentence it has a very natural descriptive force: it describes the baby as being naked while the action happens.

So infantem nudum lavat means she washes the baby naked, or more naturally in English, she washes the naked baby.

Latin often uses an adjective this way without needing extra words like while or who is.

Mater is a third-declension noun, not a first-declension one.

Its dictionary form is mater, matris.

Even though it refers to a female person, it does not follow the common -a pattern of many feminine nouns such as puella. Latin gender and declension type are related, but they are not the same thing.

So:

• mater = nominative singular
• matris = genitive singular

Yes. Depending on context, balneum can refer to:

• a bath
• a bathing place
• a bathroom
• sometimes a bathhouse

So in balneo could mean:

• in the bath
• in the bathroom
• at the baths

In a simple teaching sentence like this, in the bath or in the bathroom are both possible, depending on the intended meaning already given.