Algebra → Linear Algebra · Lesson 9

Determinants & Eigenvalues

The determinant tells you how a matrix scales area/volume. Eigenvalues and eigenvectors reveal the fundamental directions a transformation stretches or compresses — critical in data science, physics, and dynamic systems.

Key Concepts

Determinant

For 2×2: det([[a,b],[c,d]]) = ad − bc. Geometric meaning: the signed area of the parallelogram formed by the columns. det = 0 means the matrix is singular (non-invertible).

Inverse Matrix

If det(A) ≠ 0, A⁻¹ = (1/det(A)) · adj(A). For 2×2: A⁻¹ = (1/(ad−bc)) · [[d,−b],[−c,a]]. Check: AA⁻¹ = I.

Eigenvalues & Eigenvectors

Av = λv. Vector v is an eigenvector if A just scales it (by eigenvalue λ). Find λ by solving det(A − λI) = 0 (characteristic polynomial). Then find v for each λ.

Applications of Eigenvalues

Principal Component Analysis (PCA) uses eigenvectors. PageRank uses eigenvalues. Differential equations use them. Quantum mechanics: energy levels are eigenvalues of Hamiltonians.

Live Python Practice

Interactive Lab

Matrix [[a,b],[c,d]]:

Colored arrows = eigenvectors (scaled by eigenvalue shown as dashed). Gray lines = how random vectors transform. Try a diagonal matrix for clear eigenvectors.

Check Your Understanding

det([[3,2],[1,4]]) =

An eigenvector v of A satisfies:

If det(A) = 0, then A is:

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