Find the characteristic polynomial, eigenvalues, trace, and determinant of a 2×2 or 3×3 matrix with step-by-step computation and eigenvalue visualization.
The characteristic polynomial of a square matrix A is defined as det(A − λI), where λ is a variable and I is the identity matrix. The roots of this polynomial are the eigenvalues of A — the special scalars for which the equation Av = λv has non-trivial solutions.
For a 2×2 matrix, the characteristic polynomial is always a quadratic: λ² − (trace)λ + det = 0, and its roots can be found via the quadratic formula. For a 3×3 matrix, you get a cubic polynomial whose coefficients involve the trace, the sum of 2×2 principal minors, and the determinant. Solving a cubic is more involved but this calculator handles it automatically.
Computing the characteristic polynomial is central to linear algebra and its applications in physics, engineering, data science, and differential equations. Eigenvalues determine the stability of dynamical systems, the principal components in data analysis, the natural frequencies of vibrating structures, and much more. This tool lets you enter any 2×2 or 3×3 matrix and instantly see its characteristic polynomial in expanded form, all eigenvalues (real and complex), the trace, determinant, and a step-by-step breakdown of the computation.
Characteristic Polynomial Calculator helps you solve characteristic polynomial problems quickly while keeping each step transparent. Instead of redoing long algebra by hand, you can enter your inputs once and immediately inspect Characteristic Polynomial, Eigenvalues, Trace to validate your work.
This is useful for homework checks, classroom examples, and practical what-if analysis. You keep the conceptual understanding while reducing arithmetic mistakes in multi-step calculations.
2×2: p(λ) = λ² − (a+d)λ + (ad−bc); 3×3: p(λ) = −λ³ + tr(A)λ² − (sum of 2×2 minors)λ + det(A)
Result: Characteristic Polynomial shown by the calculator
Using the preset "2 × 2", the calculator evaluates the characteristic polynomial setup, applies the selected algebra rules, and reports Characteristic Polynomial with supporting checks so you can verify each transformation.
This calculator takes the problem inputs and applies the relevant characteristic polynomial relationships from your chosen method. It returns both final and intermediate values so you can audit the process instead of treating it as a black box.
Start with the primary output, then use Characteristic Polynomial, Eigenvalues, Trace, Determinant to confirm signs, magnitude, and internal consistency. If anything looks off, change one input and compare the updated outputs to isolate the issue quickly.
A strong workflow is manual solve first, calculator verify second. Repeating that loop improves speed and accuracy because you learn to spot common setup errors before they cost points on multi-step algebra problems.
It is det(A − λI), a polynomial in λ whose roots are the eigenvalues of the matrix A. Use this as a practical reminder before finalizing the result.
Eigenvalues describe fundamental properties of a linear transformation: stretching factors, stability of systems, and principal directions. They are used in PCA, differential equations, quantum mechanics, and more.
Yes. A real matrix can have complex eigenvalues, which always appear in conjugate pairs (a ± bi). This calculator shows both real and complex eigenvalues.
The trace (sum of diagonal elements) equals the sum of all eigenvalues, and the determinant equals their product. Keep this note short and outcome-focused for reuse.
This calculator supports 2×2 and 3×3 matrices with real number entries. For larger matrices, numerical methods like the QR algorithm are typically needed.
It states that every square matrix satisfies its own characteristic polynomial. If p(λ) is the characteristic polynomial of A, then p(A) = 0 (the zero matrix).