Which is correct in this following example, 'is' or 'are'?
The last important thing [is / are] the entry tickets.
And why?
Which is correct in this following example, 'is' or 'are'?
The last important thing [is / are] the entry tickets.
And why?
You can use 'is' in this sentence, because "the last important thing" is the singular subject:
The last important thing is the entry tickets.
On the other hand, if the subject were to be switched with the object subject complement* — the entry tickets — I would use 'are':
The entry tickets are the last important thing.
* Thank you, Seowjooheng!
The last important thing ... the entry tickets.
I add to what @Joachim has said.
The verb should be is, as the subject, The last important thing, is singular.
the entry tickets, however, is not object of the verb.
This is a NP1 + linking verb + NP1 construction where the entry tickets is the subject complement.
Wikipedia explains subject complement:
a subject complement or predicative of the subject is a predicative expression that follows a linking verb (copula) and that complements the subject of the sentence by either (1) renaming it or (2) describing it.
On subject-verb agreement, it says
A predicate nominative does not determine the verb. When there is a difference between the number, the verb agrees with the subject.
predicate nominative refers to the subject complement the entry tickets in our case.
A ticket, or in this case an entry ticket, is a countable noun, so one may think to use plural form to denote the "last important thing", especially since there is an "s" in "tickets".
But, the above is completely unnecessary in this case, as "is" is supposedly used as "thing" is singular to describe the collective noun of "entry tickets"
The last important thing is the entry tickets.
TL;DR:
Either variants of your sentence is possible. The variant with is is more neutral than the variant with are, but the latter is certainly grammatical as well.
The last important thing is the entry tickets.
In this variant, the Subject is [the last important thing], and as it is a singular Subject, you need to use the singular verb form is. This is the neutral way of expressing this sentence.
The last important thing are the entry tickets.
In this variant, the Subject is [the entry tickets], and as it is a plural Subject, you need to use the plural verb form are. In addition, the Subject is placed into an unusual position after the verb, which places special emphasis on the last important thing.
Subject-verb agreement
The first important point is that in English, there is number agreement between the Subject and the verb of a sentence. This means that if a sentence has a plural Subject, the verb of the sentence can't have singular inflection, and vice versa. This becomes obvious in sentence pairs like the following:
[The train sg] has[sg] stopped
[The trains pl] have[pl] stopped
In your sentence, you have two potential candidates for the Subject slot: [the last important thing], which is a singular noun phrase, and [the entry tickets], which is a plural noun phrase. You can use is if and only if the Subject of your sentence is [the least important thing]. You can use are if and only if the Subject of your sentence is [the entry tickets].
Canonical word order
The second important point is that English is a language with a comparatively inflexible word order. In the vast majority of sentences, the Subject of a sentence will precede the verb. This is why the canonical word order, i.e. the default position of sentence constituents in a normal, run-of-the-mill sentence has been described as SVO: if the sentence contains a Subject and an Object, the Subject will be placed to the left of the verb, and the Object will follow the verb. This is not only true for Objects, but for all sentence constituents that are dependents of the verb. Hence, I'll use the more general term Complement when saying that English has SVC word order.
We have established earlier that if you use is in your sentence, then we know that due to Subject-verb agreement that the singular noun phrase [the last important thing] is your Subject. The plural noun phrase [the entry tickets] can't be the Subject, but it is a verbal Complement. Thus, the sentence
[The last important thing] is [the entry ticket]
therefore follows the canonical SVC word order: The Subject is to the left of the verb, and the Complement is to the right of it.
Inversion
However, English allows deviations from the canonical SVC word order, cases in which the Subject does not precede the verb. This is most obvious in the case of questions such as
What is the last important thing?
where the verb BE has to precede the Subject. Inversions like this can also occur in sentences containing a particular type of adverb. Compare the following two variants of the same sentence:
The last important thing is never the entry tickets
Never is the last important thing the entry tickets
In the second variant, the adverb never is fronted – and as a consequence, the word order has to be inverted so that the Subject now has to follow the verb, thus yielding an AVSC word order.
Yet, there is another difference between the two sentences: the second variant places a much stronger emphasis on the adverb never than the first variant. This is not accidental: As I said above, the word order of English is relatively inflexible – but occasionally, you can use a non-canonical word order for pragmatic reasons such as special emphasis.
Subject-Complement inversion
This, then gives us the explanation why The last important point are the entry tickets is also a possible variant of this sentence. If we use are instead of is, we assign Subject status to the plural noun phrase [the entry tickets]. However, this noun phrase is not in the position where we would expect it according to the canonical SVC word order – instead, we seem to find CVS word order in which the Subject is to the right of the verb, and the Complement is to the left.
This Subject-Dependent version is possible in a range of sentences. The following examples are from 16.5.1 from the Cambridge Grammar of the English language (abridged):
On the nightstand is a pinkish-reddish envelope.
Immediately recognisable is the profoundly false tenet.
Also a nice woman is our next guest.
Arrested were Nathan Johnson and his brother Victor.
Complementing the jacket is the cap, crafted of the same denim
Three days later came another eight offers
As with the inversion involving never above, these variants have a different pragmatic load than the variants of the same sentences with canonical word order: The element to the left of the verb is in a non-canonical position, and therefore, that element receives special focus.
Summing up/Second TL;DR:
As the discussion above has shown, there are two important aspects to consider. First, the Subject has to agree in number with the verb form that is chosen. Singular Subjects go with singular verbs, and Plural Subjects go with plural verbs. Second, Subjects usually precede the verb, and dependents of the verb follow it. However, there are special circumstances where it's possible to deviate from this canonical word-word order so that the position of the Subject in relation to the verb is inverted. The resulting sentences aren't pragmatically neutral, though – they highlight the element that now occupies the slot to the left of the verb where you would usually expect to find the Subject. This is why are is also possible in your sentence, but only if you want to emphasize the last important thing.