How to Use the Characteristic Polynomial Calculator

Quick answer

Choose size and definition variant, enter all matrix entries, read expanded p(λ). Solve p(λ) = 0 separately for eigenvalues.

Introduction

The calculator on our home page is built for the core task: produce p(λ) accurately without spending exam time on long cofactor chains. It runs in your browser and supports the matrix sizes most common in coursework. Open the Characteristic Polynomial Calculator and scroll to the tool panel.

This article is not a substitute for understanding determinants. It is a verification and speed tool once you know what p(λ) means.

Read how the calculator fits into the larger eigenvalue workflow: polynomial from the tool, roots from algebra, eigenvectors from linear systems.

Tool page features and workflow

Matrix input fields accept integers, decimals, and simple fractions. Row letters with subscripts (a₁, b₂, …) match textbook layouts and reduce position confusion on 3×3 and 4×4 grids.

Determinant calculation is symbolic. The tool keeps λ as a symbol through expansion and returns a simplified polynomial rather than a number.

Instant polynomial output updates when every cell is filled. Partial grids intentionally do not show a final polynomial because incomplete matrices are not valid inputs.

Definition variant toggles let you match det(A − λI) or det(λI − A). If you are learning the determinant definition for the first time, read what a characteristic polynomial is before relying on the tool alone.

Eigenvalue support is indirect: you solve p(λ) = 0 after copying the polynomial. The calculator does not factor or list eigenvalues automatically, which keeps the focus on the characteristic polynomial step.

For manual practice between calculator checks, use the step-by-step expansion method on paper once per study session.

What the calculator implements

• p(λ) = det(A − λI) or det(λI − A)
• Output: expanded p(λ)
• No server storage of matrix entries

Implementation follows cofactor-style symbolic determinant expansion sized for n ≤ 4. Larger matrices are outside the scope of this site.

Results are formatted for readability in λ. Copy the expression into homework or a computer algebra system for factoring.

Privacy: arithmetic stays local to your browser session.

Step-by-step overview

1. Open the home page tool. Use the hero button or scroll to the section with id calculator.
2. Select matrix size. 2×2, 3×3, or 4×4 depending on the assignment.
3. Pick definition variant. Match det(A − λI) or det(λI − A) to your textbook.
4. Enter every entry. Empty cells block output. Use fractions like 3/4 when needed.
5. Read and copy p(λ). Use the polynomial for factoring practice or equation setup.
6. Solve p(λ) = 0 separately. Eigenvalues come from roots, not from the calculator panel alone.

Sample calculator session

Enter A = [[2, 1], [1, 2]] with det(A − λI). Expect p(λ) = λ² − 4λ + 3.

Change to a 3×3 diagonal matrix to see how quickly product structure appears before expansion.

Try det(λI − A) on the same matrix and observe how the polynomial may differ by an overall sign while roots stay the same.

After copying p(λ), factor by hand and compare roots with what you expect from symmetry arguments.