The form -(으)ㄹ까요 has plenty of uses so it is a great form to know. It has three major uses in total. The first way how you can use this form is to suggest to do together with someone. Secondly, you can use it is to ask for the opinion of someone else on something. Lastly, you can also use it to express one of your suppositions on something in the form of a question. As you can see it is a rather flexible form that is quite useful.
Suggestion And Opinion
When you want to make a suggestion you can use the form -(으)ㄹ까요. This form you use to ask someone to do something together. Therefore when you use this form the subject of the sentence is always 우리 (= we) and thus simply omitted. The answer to this question generally uses the present tense or the form -(으)ㅂ시다 can be used.
가: 주말에 같이 영화를 볼까요?
나: 네, 좋아요. 같이 봐요.
A: Shall we watch a movie together during the weekend?
B: Yes, sounds great. Let’s watch one.
However, you can also use the form -(으)ㄹ까요 to ask for someone’s thoughts or opinion on something. The subject of the question is normally 저 (= I) and thus simply omitted from the sentence. The reply to this question generally uses the imperative forms -(으)세요 or its negative -지 마세요.
가: 더운 것 같아요. 창문을 열까요?
나: 오늘 먼지가 많으니까 열지 마세요.
A: It feels a bit hot. Shall I open a window?
B: Today there is a lot of dust so don’t open it.
Conjecture And Supposition
Another use of the form -(으)ㄹ까요 is to express your supposition or conjecture in the form of a question rather than a statement. This is a great way to invite someone else to tell you what he thinks. The person responding to your question will most likely use the form -(으)ㄹ 거예요 when he is certain, and -(으)ㄴ/는 것 같다 when is less certain.
가: 민수 씨가 이 책을 좋아할까요?
나: 네, 작가를 좋아해서 좋을 거예요.
A: Do you think Minsu will like this book?
B: Yes, he likes the author, so it will be great.
How To Use
The form -(으)ㄹ까요 is not difficult to use, but there are some differences between using it for expressing supposition; or using it to make suggestions or asking for opinions. These differences are important to keep in mind. However, the conjugation rules are the same for all of its uses. The form simply follows the ‘batchim/no batchim’ rule. The only thing you need to keep in mind is that verbs, whose stem end in a ㄹ, are treated as if the verb ends with no batchim. Lastly, keep the irregular verbs in mind.
Now to grammatical differences between the uses. When you use the form -(으)ㄹ까요 to make a suggestion or ask for someone’s opinion, you can only use it with verbs, not adjectives. Furthermore, you cannot use the form in the past tense as it would make little to no sense.
On the other hand, when you use the form -(으)ㄹ까요 to express your supposition in the form of a question, you can use it with both verbs and adjectives. And, you can use it in the past tense. The past tense of this form is -았/었을까요.