Characteristic Polynomial Calculator

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Choose whether to work with a 2x2, 3x3, or 4x4 square matrix.
Entry in row 1, column 1.
Entry in row 1, column 2.
Entry in row 2, column 1.
Entry in row 2, column 2.
Characteristic polynomial
lambda^2 - 5lambda - 6

p(lambda) = det(A - lambdaI), expressed as a polynomial in lambda

Trace (sum of diagonal)5
Determinant-6
Real eigenvalueslambda1 = 6, lambda2 = -1
Number of real eigenvalues2
Trace5
Determinant-6
Real eigenvalues2

Characteristic polynomial: p(lambda) = lambda^2 - 5lambda - 6

  • The trace of this matrix is 5, which equals the sum of all eigenvalues (counted with multiplicity).
  • The determinant is -6, which equals the product of all eigenvalues. A non-zero determinant confirms the matrix is invertible.
  • All 2 eigenvalues are real. If they are all distinct, the matrix is diagonalizable.
  • For a 2x2 matrix, p(lambda) = lambda^2 - tr(A)*lambda + det(A) -- knowing just trace and determinant is enough to write the polynomial.

Next stepSet p(lambda) = 0 and solve to find the eigenvalues. For each eigenvalue, solve (A - lambdaI)v = 0 to find the corresponding eigenvectors.

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