Characteristic Polynomial
Compute the characteristic polynomial, trace, determinant, and eigenvalues of a 2x2 matrix instantly. Free online linear algebra tool with step-by-step results for students and professionals.
About This Calculator
The characteristic polynomial of a square matrix is a fundamental invariant in linear algebra. For a 2x2 matrix A = [[a, b], [c, d]], the characteristic polynomial is p(lambda) = lambda^2 - tr(A)lambda + det(A), where tr(A) = a + d is the trace and det(A) = ad - bc is the determinant.
The roots of this quadratic polynomial are the eigenvalues of the matrix, given by lambda = (tr(A) +/- sqrt(tr(A)^2 - 4det(A))) / 2. Eigenvalues reveal whether the matrix compresses, stretches, or rotates vectors in the plane and are essential in differential equations, quantum mechanics, data science (PCA), and engineering stability analysis.
How to use: Enter the four entries a, b, c, d of your 2x2 matrix, then click Compute. The tool displays the characteristic polynomial, trace, determinant, and eigenvalues (real or complex) instantly.
Frequently Asked Questions
What is the characteristic polynomial of a 2x2 matrix?
For a 2x2 matrix A = [[a, b], [c, d]], the characteristic polynomial is lambda^2 - tr(A)lambda + det(A), where tr(A) = a + d is the trace and det(A) = ad - bc is the determinant. The roots of this polynomial are the eigenvalues of the matrix.
How do you find eigenvalues from the characteristic polynomial?
Eigenvalues are the roots of the characteristic polynomial lambda^2 - tr(A)lambda + det(A) = 0. For a 2x2 matrix, they are given by lambda = (tr(A) +/- sqrt(tr(A)^2 - 4det(A))) / 2. If the discriminant is positive, you get two distinct real eigenvalues; if zero, one repeated real eigenvalue; if negative, a complex conjugate pair.
What does the characteristic polynomial tell you about a matrix?
The characteristic polynomial encodes key invariants of the matrix: its trace (sum of eigenvalues), determinant (product of eigenvalues), and eigenvalues themselves. It determines whether the matrix is diagonalizable, invertible (det ≠ 0), and provides insight into its geometric transformations.
Is this characteristic polynomial calculator free?
Yes, all calculators on Calculy are completely free to use with no registration or hidden fees.
What is the difference between trace and determinant?
The trace is the sum of the diagonal entries (a + d) and equals the sum of the eigenvalues. The determinant is ad - bc and equals the product of the eigenvalues. A zero determinant means the matrix is singular (non-invertible).
Can I use this calculator for 3x3 matrices?
This calculator currently supports 2x2 matrices only. For a 3x3 matrix, the characteristic polynomial is a cubic polynomial lambda^3 - tr(A)lambda^2 + ... - det(A), which requires solving a cubic equation -- a more complex computation not yet available on this tool.