Good resources on matrix equations

The first reference listed in Wikipedia has a table of contents which looks appropriate:

Gohberg, Israel; Lancaster, Peter; Rodman, Leiba (2009) [1982]. Matrix Polynomials. Classics in Applied Mathematics.

But as I commented above, you can do it for yourself if you think about the fact that $A$ and $P^{-1}AP$ will always be solutions to the same matrix polynomials. Then solve the equation for the case where $A$ is in Jordan form. All other solutions will be conjugates of these type of solutions. (There will often be several Jordan forms which solve the equation, depending on multiplicity of roots, etc.)

Take a look at the Naval Academy course on Matrix Theory. It emphasizes matrix manipulation more than proofs shmoofs. And is not a general LA class.

https://www.usna.edu/Users/math/hottovy/Teaching/SM261.php

There are some worksheets and notes you could look at also on the SM261 page.

Here is a link to Hefferon's LA text (used for the 261 class), which is progressive and FREE and emphasizes matrix manipulation at the start.

http://joshua.smcvt.edu/linearalgebra/

Other than that, I heartily recommend the chapter on matrices in Kreyszig's Advanced Engineering Mathematics. It is very self-standing and gives you exactly what you want (learning to rock the matrices around the clock, not abstraction for the proof lovers). It's not free, but you can get used copies (edition not important, I liked the 5th). About 50% of the problems had answers in the back, so helpful for drill. Also, you can just look at in the library (maybe even zox the chapter).

https://www.thriftbooks.com/w/advanced-engineering-mathematics-8th-edition_erwin-kreyszig/263019/item/2112180/?mkwid=%7cdc&pcrid=395931773744&pkw=&pmt=&slid=&plc=&pgrid=80257881302&ptaid=pla-838159019236&gclid=EAIaIQobChMIvojsuOLF6gIVDbLICh29-QcgEAQYAiABEgL2FfD_BwE#isbn=0471862517&idiq=2112180

You might also look for a cheap used copy of Kolman Linear Algebra with Applications (I have third edition). It emphasizes matrix manipulation much more than abstraction. And gets right into it, versus building theory first. It looks like the recent editions are more abstract and there is a second author (normal scope creep that you see for commercial reasons as books age: appeals to professors who already know the stuff and sniff at ommissions and they select the books, but can be frustrating for students or self studiers). But looking at my third edition on the shelf is very user friendly.