Day 1 — Linear Algebra
Time: ~45 min · Topic: linear algebra (matrices, eigenvalues, vector spaces)
The problem
Let A be a 3×3 matrix with tr(A) = 6, tr(A²) = 14, and det(A) = 6. Find the eigenvalues of A.
Hint: The trace and determinant encode symmetric functions of the eigenvalues. Can you recover the missing piece — the sum of products of pairs?
Try it yourself
Take a few minutes. Don't peek at the steps. If you get stuck, the hint above is enough to nudge you forward.
Step-by-step solution
- Recall that for a matrix A with eigenvalues λ₁, λ₂, λ₃: tr(A) = λ₁ + λ₂ + λ₃ = 6, tr(A²) = λ₁² + λ₂² + λ₃² = 14, and det(A) = λ₁λ₂λ₃ = 6.
- Use the identity (λ₁ + λ₂ + λ₃)² = λ₁² + λ₂² + λ₃² + 2(λ₁λ₂ + λ₁λ₃ + λ₂λ₃). Substituting: 36 = 14 + 2(λ₁λ₂ + λ₁λ₃ + λ₂λ₃), so λ₁λ₂ + λ₁λ₃ + λ₂λ₃ = 11.
- By Vieta's formulas, the eigenvalues are roots of the characteristic polynomial: λ³ − 6λ² + 11λ − 6 = 0.
- Factor: (λ − 1)(λ − 2)(λ − 3) = 0. The eigenvalues are λ = 1, 2, and 3.
Answer: The eigenvalues of A are 1, 2, and 3.
Why this matters
Google's original PageRank algorithm worked by finding the dominant eigenvalue of the entire internet's link matrix — the same eigenvalue theory at play in this problem, just scaled to billions of dimensions.
This day in math: The Man Who Mapped Magnetic Fields
April 21, 1774 · Jean-Baptiste Biot
On April 21, 1774, French mathematician and physicist Jean-Baptiste Biot was born in Paris. In 1820, Biot and Félix Savart discovered the mathematical law governing how electric currents create magnetic fields — now known as the Biot-Savart Law. He also pioneered the study of light polarization through organic substances, establishing Biot's Law of Rotary Polarization.
Why it matters: The Biot-Savart Law is foundational to electromagnetism and is used today in everything from MRI machines to electric motor design. His work on light polarization laid the groundwork for modern optics and even helps chemists identify molecular structures.
Did you know?
Cicadas have evolved to emerge on prime number cycles — every 13 or 17 years — and mathematicians believe this is no coincidence. By syncing their life cycles to prime numbers, these insects gain a powerful survival advantage that's built on pure number theory.
Because prime numbers have no smaller divisors other than 1, a predator with a repeating 2-, 3-, 4-, or 5-year cycle will almost never sync up with a 13- or 17-year emergence. For example, a predator on a 4-year cycle would only overlap with a 17-year cicada once every 68 years (4 × 17). If cicadas emerged every 12 years instead, predators on 2-, 3-, 4-, or 6-year cycles would ALL line up repeatedly. Primes minimize these dangerous overlaps — evolution discovered number theory millions of years before we did.
Done?
- Solved the problem (or read through the steps)
- Read the history note
- Read the "did you know" fact