Why Do Determinants Count Trees?

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⬣ ABOUT ⬣
An immensely satisfying theorem that allows us to calculate a graph's number of spanning trees using matrix determinants.

⬣ TIMESTAMPS ⬣
00:00 - Intro
01:45 - Initial Attempts and Plan
04:02 - A Short History of Determinants
07:46 - Determinants and Permutations
12:14 - From Spanning Trees to Arborescences
17:00 - Counting Arborescences
24:08 - Final Example
27:00 - Concluding Thoughts

⬣ ANSWERS ⬣
For the "Extra Practice" matrices, we should get these determinants:
1. (the 3x3 matrix) 2
2. (the 4x4 matrix) 9

⬣ INVESTIGATORS ⬣
Nothing for you here. Sorry!

⬣ REFERENCES ⬣
James Joseph Sylvester Memorial Address - Fabian Franklin
https://www.ams.org/journals/bull/189...

The collected mathematical papers of James Joseph Sylvester
https://archive.org/details/collected...

"The determinant | Chapter 6, Essence of linear algebra" - 3Blue1Brown
   • The determinant | Chapter 6, Essence of li...  

"Interpreting determinants in terms of area | Matrices | Precalculus | Khan Academy" - Khan Academy
   • Interpreting determinants in terms of area...  

"What is a determinant?" - Leios Labs
   • What is a determinant?  

"Why is the determinant like that?" - broke math student
   • Why is the determinant like that?  

⬣ CREDITS ⬣
Music by Danijel Zambo, Sky Toes & Guilio Fazio