Uploaded September 2019 | Updated September 2026, 1 week ago
2019 AIME I Problem 1. There's a fairly simple solution, but what would we do if we didn't figure it out?
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2019 AIME I Problem 1. There's a fairly simple solution, but what would we do if we didn't figure it out?
New math videos every Monday and Friday. Subscribe to make sure you see them!
![How the Matrix Characteristic Polynomial is Connected to F[x]-Modules](https://i.ytimg.com/vi/jCt6mR3QtPk/mqdefault.jpg "How the Matrix Characteristic Polynomial is Connected to F[x]-Modules
Rational canonical form: https://youtu.be/q5uj4o0O5R0
Similar matrices isomorphism: https://youtu.be/-ligAAxFM8Y
Intro to F[x]-modules: https://youtu.be/H44q_Urmts0
The characteristic polynomial described by the determinant det(xI-A) is very useful when studying matrices. It turns out that its related to modules as well! Here we show how the characteristic polynomial is related to the module form of an arbitrary matrix.
Ring & Module Theory playlist: https://www.youtube.com/playlist?list=PLug5ZIRrShJExMapwnaKTFXDYbKeWDXq7
0:00 Computing the determinant
10:07 Similar matrices have same polynomial
12:40 Proof for a general matrix
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Music: C418 - Pr Department")
