Linear Algebra Intermediate

Eigenvalue & Matrix Solver

Enter any n×n matrix (up to 50×50) — get determinant, row reduction steps, upper triangular form, eigenvalues, and characteristic matrix analysis.

Supports up to 50×50 matrices

QR iteration for eigenvalues

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Enter matrix values and press Solve to compute the determinant by triangular form

01 Original Matrix A

02 Row Reduction Steps

Show/Hide Elimination Steps ▶

03 Upper Triangular Form U

Diagonal entries:

04 Determinant

det(A) = × (product of diagonal entries of U)

det(A) =

05 Eigenvalues & Eigenvectors Numerical (Power Iteration)

06 Characteristic Matrix A − λI & Symbolic Upper Triangular Form

07 Identity Matrix · A = A

Eigenvalues & Determinants

det(A) = (−1)^(swaps) × u₁₁ × u₂₂ × ⋯ × uₙₙ

The determinant equals the product of all eigenvalues: det(A) = λ₁ · λ₂ · … · λₙ. This calculator finds it via row reduction to upper triangular form, then multiplies the diagonal, adjusting sign for row swaps.

Eigenvalues satisfy det(A − λI) = 0. For matrices ≤ 5×5 this tool uses QR iteration to find them numerically.

If det(A) = 0, exactly one eigenvalue is 0, meaning A is singular (not invertible) and Ax = 0 has infinitely many solutions.

When To Use This Tool

Computing the determinant of any square matrix
Finding eigenvalues via QR iteration
Checking invertibility (det ≠ 0)
Understanding spectral decomposition
Principal component analysis (PCA) foundations

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