Eigenvalue Calculator
Solve det(A − λI) = 0 for 2×2 and 3×3 matrices and see the characteristic polynomial factored step by step with the exact real eigenvalues - live as you type.
Exact real eigenvalues are shown, including square roots. Complex roots are not currently shown.
Your matrix
Characteristic polynomial
Eigenvalues
Solution method
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How to use the eigenvalue calculator
- Pick 2×2 or 3×3 and type a number (decimals and fractions like 1/5 both work) into every cell.
- Read the characteristic polynomial: the determinant of A − λI, expanded and collected in λ.
- Rational eigenvalues are shown exactly with multiplicity; entries and fractions stay exact throughout.
Want the vectors too? The eigenvector calculator uses the same matrix and pairs with this page.
What eigenvalues tell you
Eigenvalues summarize a matrix at a glance: their product is the determinant, their sum is the trace, and a zero eigenvalue means the matrix is singular with no inverse. The original PageRank ranked pages by the dominant eigenvector of the link matrix, and engineers use the largest eigenvalue to test whether bridges and circuits stay stable. With integer entries, rational eigenvalues appear here as exact fractions, not rounded decimals.