Следовательница думает, что у каждой подозреваемой должен быть мотив.

Следовательница is the explicitly feminine form, meaning female investigator.
Следователь is the standard dictionary form and is often used for a person of either sex in official contexts, but следовательница clearly tells you the investigator is a woman.

So:

• следователь = investigator
• следовательница = female investigator

A learner may also notice that feminine profession nouns in Russian can vary in how common or stylistically neutral they sound. In this sentence, следовательница is perfectly understandable and highlights the woman’s gender.

Because it is originally an adjective, but here it is being used as a noun. This is very common in Russian.

The full phrase behind it would be something like:

• подозреваемая женщина = a suspected woman / female suspect

But Russian often drops the noun when it is obvious, so подозреваемая by itself means female suspect.

In your sentence, the form is подозреваемой because the grammar requires a different case, but it still means suspect.

Russian often expresses possession with у + genitive, not with a verb equivalent to English have.

So instead of saying:

• Each suspect should have a motive

Russian structures it more like:

• At each suspect, there should be a motive

That is why you get:

• у каждой подозреваемой = for each female suspect / with each female suspect / each female suspect has

This is the normal Russian way to say that someone has something.

It is genitive singular feminine.

That happens because the preposition у requires the genitive case.

So:

• каждая подозреваемая = nominative, basic form
• у каждой подозреваемой = genitive after у

Both words change because they agree with each other:

• каждой matches
• подозреваемой matches

They are both feminine singular genitive.

Because Russian uses the singular after каждый when talking about people or things one by one.

So:

• каждая подозреваемая = each female suspect
• у каждой подозреваемой = of each female suspect / for each female suspect

This is different from English, where each can feel semantically plural, but grammatically Russian treats it as singular.

Because должен agrees with мотив, not with подозреваемой.

This is a very common point of confusion. In this sentence, the thing that should be is мотив.

The structure is basically:

• мотив должен быть = the motive should be

And then:

• у каждой подозреваемой = with each suspect / each suspect should have

So the masculine form должен is used because:

• мотив is masculine

If the noun were feminine, you would use должна:

• У каждой подозреваемой должна быть причина.

Мотив is in the nominative case.

That is because it is the grammatical subject of the clause:

• мотив должен быть = a motive should be

Even though English says each suspect should have a motive, Russian structures it so that motive is the subject, while the possessor is expressed with у + genitive.

Because it follows должен.

In Russian, after words like должен, может, хочет, and similar verbs or predicative forms, the next verb normally appears in the infinitive.

So:

• должен быть = should be / must be

This works just like English must be, where be is also in a basic form.

Что introduces a subordinate clause, like English that.

So:

• Следовательница думает = The investigator thinks
• что у каждой подозреваемой должен быть мотив = that each suspect should have a motive

Russian often uses что in exactly this way after verbs of thinking, saying, knowing, believing, and so on.

It can suggest either one depending on context.

Должен often expresses:

• obligation
• expectation
• logical necessity

In this sentence, it most naturally means something like:

• should have
• ought to have
• possibly must have

A detective talking this way may mean: logically, every suspect is expected to have a motive.

So the exact English wording depends on tone, but the Russian grammar stays the same.

Because they are both:

• feminine
• singular
• genitive

Russian adjectives and adjective-like words must agree with the noun they describe.

Here:

• каждой = of each
• подозреваемой = suspect

They match in gender, number, and case, so they take the same ending pattern.

Yes. Russian word order is flexible, though the emphasis changes.

The original:

• Следовательница думает, что у каждой подозреваемой должен быть мотив.

This is a very natural neutral order.

You could also say:

• Следовательница думает, что мотив должен быть у каждой подозреваемой.

That puts more focus on мотив.

Or:

• У каждой подозреваемой должен быть мотив, думает следовательница.

That sounds more literary or stylistically marked.

So the original version is a normal, clear way to say it.

Yes. The sentence is clearly talking about women:

• следовательница = female investigator
• подозреваемой = female suspect

Russian marks gender much more openly than English does in many nouns and adjective forms. So this sentence does not just mean an investigator thinks each suspect should have a motive in a gender-neutral sense. It specifically refers to:

• a woman investigator
• female suspects

If the suspects were male, you would get:

• у каждого подозреваемого должен быть мотив

If the investigator were referred to with the masculine/common form, you might see:

• Следователь думает...