Is it necessary to begin a new paragraph after a person speaks?

When I was in school, I was taught to always begin a new paragraph after a quotation or after a speaker concludes a segment of dialogue. However, in recent years, I have seen authors abandon this rule, continuing a scene's description after a quotation. To clarify, I'm not asking whether I should create a new paragraph when changing speakers. Rather, I'm asking if a paragraph such as:

Joe said, "This is a placeholder sentence." He wrote the same statement on the chalkboard.

is correct as is, or if the second sentence should begin a second paragraph.

Solution 1:

This is purely a matter of style. Even Dickens' publishers didn't follow that rule. This is from Google Books' reproduction of A Tale of Two Cities published in Philadelphia in 1859:

However, at least one reproduction of this passage does split this single paragraph between the end of the quote and And.

Paragraphs generally hold a single idea. If there is narration which follows directly from the quotation and it could be understood as part of the same idea, it's reasonable for it to be all in the same paragraph.