How do you find the characteristic polynomial of a 2x2 matrix?

For a 2x2 matrix with entries a, b, c, d, the characteristic polynomial is lambda squared minus (a+d) lambda plus (ad minus bc). This equals lambda squared minus trace times lambda plus determinant.

เครื่องคำนวณพหุนามลักษณะเฉพาะ

คำนวณพหุนามลักษณะเฉพาะ det(A − λI) ของเมทริกซ์จัตุรัส รองรับเมทริกซ์ขนาด 2×2 ถึง 6×6 พร้อมการกระจายโคแฟกเตอร์ทีละขั้นตอน การหาค่าคงตัวเฉพาะ การวิเคราะห์สัมประสิทธิ์ และการแสดงภาพพหุนามแบบโต้ตอบ

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The Characteristic Polynomial Calculator computes the characteristic polynomial $p(\lambda) = \det(\lambda I - A)$ of any square matrix from 2×2 to 6×6. Enter your matrix values, and instantly get the polynomial in both expanded and factored form, eigenvalues with multiplicities, a coefficient analysis table, an interactive polynomial graph, and a complete step-by-step solution with MathJax-rendered formulas.

What Is the Characteristic Polynomial?

The characteristic polynomial of an $n \times n$ matrix $A$ is defined as:

$$p(\lambda) = \det(\lambda I - A)$$

This is a degree-$n$ polynomial in $\lambda$, and its roots are exactly the eigenvalues of $A$. The characteristic polynomial encodes fundamental invariants of the matrix: its trace equals the negative of the $\lambda^{n-1}$ coefficient, and its determinant equals the constant term (up to sign). By the Cayley–Hamilton theorem, every square matrix satisfies its own characteristic equation: $p(A) = 0$.

Key Concepts

🔢

Eigenvalues

Roots of p(λ) = 0. These are the values λ where det(A − λI) = 0.

∑

Trace = Σλᵢ

Sum of diagonal entries equals sum of all eigenvalues.

∏

Det = ∏λᵢ

Determinant equals the product of all eigenvalues.

📜

Cayley–Hamilton

Every matrix satisfies its own characteristic equation: p(A) = 0.

Characteristic Polynomial Formulas by Size

Size	Characteristic Polynomial p(λ)	Key Properties
2×2	$\lambda^2 - \text{tr}(A)\lambda + \det(A)$	Always degree 2; two roots (real or complex conjugate pair)
3×3	$\lambda^3 - \text{tr}(A)\lambda^2 + (\text{sum of 2×2 minors})\lambda - \det(A)$	At least one real root guaranteed
n×n	$\det(\lambda I - A) = \lambda^n - s_1\lambda^{n-1} + s_2\lambda^{n-2} - \ldots$	$s_k$ = sum of all k×k principal minors

Applications of the Characteristic Polynomial

Field	Application	How the Characteristic Polynomial Helps
Differential Equations	Solving linear ODE systems	Eigenvalues from p(λ) determine solution modes (growth, decay, oscillation)
Control Theory	System stability analysis	Roots of the characteristic polynomial indicate stable vs unstable modes
Quantum Mechanics	Energy levels of systems	Eigenvalues of Hamiltonian matrix are measurable energy states
Graph Theory	Spectral graph analysis	Characteristic polynomial of adjacency matrix encodes graph structure
Vibration Analysis	Natural frequencies	Eigenvalues give resonant frequencies of mechanical systems
Data Science	PCA / dimensionality reduction	Largest eigenvalues identify principal components in covariance matrices

How to Use the Characteristic Polynomial Calculator

Choose matrix size: Use the +/− buttons to select a matrix from 2×2 to 6×6. Or click a quick example to load a preset matrix.
Enter matrix values: Type numbers into the matrix grid. Use Tab or arrow keys to navigate between cells. The diagonal cells are highlighted in blue to help with orientation.
Click Calculate: The calculator forms the matrix (A − λI), computes the determinant symbolically to produce the characteristic polynomial, then factors it to find eigenvalues.
Review the results: Examine the characteristic polynomial in expanded and factored forms. Check the eigenvalue cards for roots and multiplicities. The interactive graph shows where p(λ) crosses zero.
Explore step-by-step: Use the step navigator or Auto button to walk through the complete derivation — from forming A − λI to the final verification via trace and determinant.

FAQ

What is a characteristic polynomial?

The characteristic polynomial of a square matrix A is p(λ) = det(λI − A), a degree-n polynomial whose roots are the eigenvalues of A. It encodes essential information about the matrix including its eigenvalues, trace, and determinant. The characteristic polynomial is one of the most fundamental objects in linear algebra.

How do you find the characteristic polynomial of a 2×2 matrix?

For a 2×2 matrix [[a, b], [c, d]], the characteristic polynomial is λ² − (a+d)λ + (ad − bc). This simplifies to λ² − tr(A)λ + det(A), where tr(A) = a+d is the trace and det(A) = ad − bc is the determinant. The two roots give you the eigenvalues.

What is the relationship between the characteristic polynomial and eigenvalues?

The eigenvalues of a matrix are exactly the roots of its characteristic polynomial. If λ₀ is a root of p(λ) = det(λI − A) = 0, then λ₀ is an eigenvalue. The algebraic multiplicity of an eigenvalue is its multiplicity as a root of p(λ). For example, if p(λ) = (λ − 3)²(λ − 1), then λ = 3 has algebraic multiplicity 2 and λ = 1 has algebraic multiplicity 1.

Can a characteristic polynomial have complex roots?

Yes. Even for a real matrix, the characteristic polynomial can have complex roots (eigenvalues). Complex eigenvalues of real matrices always come in conjugate pairs: if a + bi is an eigenvalue, then a − bi is also an eigenvalue. For example, the rotation matrix [[0, −1], [1, 0]] has characteristic polynomial λ² + 1, with roots ±i.

What do the coefficients of the characteristic polynomial tell us?

The coefficients encode important matrix invariants. The leading coefficient is always 1 (monic polynomial). The coefficient of λ^(n−1) equals −tr(A) (negative trace). The constant term equals (−1)ⁿ det(A). More generally, the coefficient of λ^(n−k) is (−1)^k times the sum of all k×k principal minors of A. These are called the elementary symmetric polynomials of the eigenvalues.

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by miniwebtool team. Updated: 2026-04-13

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What Is the Characteristic Polynomial?

Key Concepts

Characteristic Polynomial Formulas by Size

Applications of the Characteristic Polynomial

How to Use the Characteristic Polynomial Calculator

FAQ

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เครื่องมือเด่น:

Size	Characteristic Polynomial p(λ)	Key Properties
2×2	\(\lambda^2 - \text{tr}(A)\lambda + \det(A)\)	Always degree 2; two roots (real or complex conjugate pair)
3×3	\(\lambda^3 - \text{tr}(A)\lambda^2 + (\text{sum of 2×2 minors})\lambda - \det(A)\)	At least one real root guaranteed
n×n	\(\det(\lambda I - A) = \lambda^n - s_1\lambda^{n-1} + s_2\lambda^{n-2} - \ldots\)	\(s_k\) = sum of all k×k principal minors