The 2×2 Eigenvalue Shortcut Every Student Should Know

Finding eigenvalues of a 2×2 matrix usually means a characteristic polynomial and the quadratic formula. Here's a shortcut so fast you can do it in your head: eigenvalues = mean ± distance, straight from the trace and the determinant. Plus a one-line trick for the eigenvectors! What you'll learn: Why the eigenvalues always add to the trace and multiply to the determinant The shortcut: λ = m ± √(m² − p), where m = trace/2 and p = determinant The eigenvector shortcut: v = (b, λ − a) Worked examples: [3 1; 2 2] → λ = 4, 1 | [5 2; 3 4] → λ = 7, 2 Complex eigenvalues ([1 −2; 2 1] → 1 ± 2i) and repeated eigenvalues ([3 1; −1 1] → 2, twice) The Fibonacci matrix → the golden ratio → Binet's formula Solving x' = Ax and classifying stability from trace & determinant alone Chapters: 0:00 Hook 0:24 The slow way (and why to skip it) 0:44 Two facts: trace and determinant 1:02 Derivation: mean ± distance 1:28 The shortcut formula 1:48 Example 1 2:18 The eigenvector shortcut 2:48 Example 1: eigenvectors 3:13 Example 2 3:50 Complex eigenvalues 4:26 Repeated eigenvalues 4:59 Fine print: b = 0 and the sanity check 5:27 Fibonacci and the golden ratio 6:05 Binet's formula 6:28 Differential equations 6:49 Stability from trace and determinant 7:17 Recap 7:40 Your turn: challenge 8:05 Subscribe Your turn! Find the eigenvalues AND eigenvectors of A = [7 3; 2 6]. Hint: m = 6.5 and the number under the root is a perfect square. Bonus: B = [2 −5; 1 0]. Drop your answers in the comments, and I'll pin the answer key! If this saved you a page of algebra, subscribe and give the video a like. New math tricks every week. #Eigenvalues #LinearAlgebra #MathTricks #Mathematics #CollegeMath #MathMotionLab #Matrix #StudyHacks