Why Do Determinants Count Trees?

Learn for free on Brilliant for a full 30 days: https://brilliant.org/AnotherRoof/ You’ll also get 20% off an annual Premium subscription. ⬣ LINKS ⬣ ⬡ PATREON: https://www.patreon.com/anotherroof ⬡ CHANNEL: https://www.youtube.com/c/AnotherRoof ⬡ WEBSITE: https://anotherroof.top ⬡ SUBREDDIT: https://www.reddit.com/r/anotherroof ⬡ TWITCH: https://www.twitch.tv/anotherroof ⬣ 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) 79 ⬣ INVESTIGATORS ⬣ Nothing for you here. Sorry! ⬣ REFERENCES ⬣ James Joseph Sylvester Memorial Address - Fabian Franklin https://www.ams.org/journals/bull/1897-03-09/S0002-9904-1897-00424-4/S0002-9904-1897-00424-4.pdf The collected mathematical papers of James Joseph Sylvester https://archive.org/details/collectedmathem01sylvrich/page/n125/mode/2up "The determinant | Chapter 6, Essence of linear algebra" - 3Blue1Brown https://www.youtube.com/watch?v=Ip3X9LOh2dk "Interpreting determinants in terms of area | Matrices | Precalculus | Khan Academy" - Khan Academy https://www.youtube.com/watch?v=ER_96Mx-KEE "What is a determinant?" - Leios Labs https://www.youtube.com/watch?v=vvR3JSXO2fo "Why is the determinant like that?" - broke math student https://www.youtube.com/watch?v=Sv7VseMsOQc ⬣ CREDITS ⬣ Music by Danijel Zambo, Sky Toes & Guilio Fazio