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Linear Algebra

Eigenvalues, matrix operations, row reduction, and vector spaces — exact symbolic results with every transformation shown.

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What Zeretis can solve

Zeretis handles linear algebra exactly — no floating-point rounding, no approximate eigenvalues. Results are returned in exact rational or radical form.

Eigenvalues & eigenvectors

Characteristic polynomial, exact eigenvalues, corresponding eigenvectors for any square matrix.

Matrix operations

Addition, multiplication, powers, transpose, conjugate transpose.

Determinants

Any size. Cofactor expansion, row reduction method, exact rational results.

Matrix inverse

Exact inverse via adjugate or row reduction. Detects singular matrices.

Row reduction (RREF)

Step-by-step Gaussian elimination to reduced row echelon form.

Vector spaces

Null space, column space, rank, basis, linear independence checking.

How to type linear algebra problems

Eigenvalues

eigenvalues of [[2,1],[1,2]]

→ λ₁ = 1, λ₂ = 3

Matrix multiplication

multiply [[1,2],[3,4]] by [[5,6],[7,8]]

→ [[19,22],[43,50]]

Determinant

determinant of [[1,2,3],[4,5,6],[7,8,9]]

→ 0 (matrix is singular)

Row reduction

row reduce [[2,4,6],[1,3,5],[0,1,2]]

→ Step-by-step RREF with each row operation labelled

Inverse

inverse of [[1,2],[3,4]]

→ [[-2, 1],[3/2, -1/2]]

Best practices

Enter matrices as [[row1],[row2]] — e.g. [[1,2,3],[4,5,6],[7,8,9]]
For eigenvectors, ask "eigenvectors of..." to get the full eigenvector for each eigenvalue
Row reduction shows every step — labelled R₁ ↔ R₂, R₂ − 2R₁, etc.
If a matrix is singular, the solver tells you and shows why (linearly dependent rows)
For symbolic matrices with parameters: "find eigenvalues of [[a,1],[1,a]]"
Null space: "find the null space of [[1,2,3],[4,5,6]]"

📐Algebra ∫Calculus 🔀Differential Equations ∑Series & Sequences 📏Trigonometry

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