To answer the question "Is the use of a screencap from The Matrix in this blog post fair use?"
This is clear-cut: no.
17 USC 107:
Notwithstanding the provisions of sections 106 and 106A, the fair use of a copyrighted work, including such use by reproduction in copies or phonorecords or by any other means specified by that section, for purposes such as criticism, comment, news reporting, teaching (including multiple copies for classroom use), scholarship, or research, is not an infringement of copyright. In determining whether the use made of a work in any particular case is a fair use the factors to be considered shall include—
- the purpose and character of the use, including whether such use is of a commercial nature or is for nonprofit educational purposes;
- the nature of the copyrighted work;
- the amount and substantiality of the portion used in relation to the copyrighted work as a whole; and
- the effect of the use upon the potential market for or value of the copyrighted work.
The purpose of the use: purely decorative. The text of the blog post makes no reference whatsoever to the image, and in fact, the meaning of the post would be unchanged if the image were removed. This is by far the biggest strike against it being fair use: "decoration" does not fall under any of criticism, comment, news reporting, teaching, scholarship, or research. The fact that the blog is (presumably) non-profit is irrelevant here.
The nature of the work: The Matrix is purely a work of fiction, so you can't pull out a defense such as "public interest" for your claim of fair use. On the other hand, being a work of fiction makes a "criticism" or "scholarship" defense easier.
The amount and substantiality of the portion used: the use of a single frame, and one that is not particularly plot-relevant works in favor of a claim of fair use, but also against it. Since the frame chosen doesn't reveal a plot point or anything, it doesn't reduce the value of the original work, but it also makes it harder to make a claim that the image illustrates a point of commentary or criticism.
The effect of the use on the potential market or value of the original: None. Seeing this one random screenshot can't substitute for seeing the movie. This is the biggest point in favor of a claim of fair use, but "it doesn't do any harm" pales in comparison to the problems of point 1.
If I were a lawyer contacted by the Wachowski brothers about suing the author of the blog post, my response would be "send a nastygram or a DMCA takedown, but don't waste your time with a lawsuit. The lack of damages (point 4) means you'll spend far more on lawyers than you can hope to get by winning in court."
