Characteristic Polynomial of a 3x3 Matrix

The characteristic polynomial of a 3x3 matrix calculator computes the characteristic polynomial of a 3x3 matrix.

INSTRUCTIONS: Enter the following:

• (A)  3x3 matrix

Polynomial (CP): The calculator returns the:

• Characteristic Polynomial of  matrix A
• Trace of matrix A
• Determinant of matrix A

Matrix Calculators

• Determinant of 3-by-3 Matrix
• Characteristic Polynomial of a 3x3 matrix
• Inverse of a 3x3 Matrix
• Transpose of a 3x3 Matrix
• Trace of a 3x3 Matrix
• Mirror of a 3x3 Matrix
• 3x3 Matrix Characteristics (Trace, Determinant, Inverse, Characteristic Polynomial)
• Product of a 3x3 matrix and a Scalar
• Product of a 3x3 matrix and a 3x1 matrix
• Product of two 3x3 matrices
• Solving 3 Equations with 3 Unknowns
• Cramer's Rule (three equations, solved for x, y and z)
• Cramer's Rule Calculator

The Math

The characteristic polynomial (CP) of an nxn matrix `A` is a polynomial whose roots are the eigenvalues of the matrix `A`. It is defined as `det(A-λI)`, where `I` is the identity matrix. The coefficients of the polynomial are determined by the determinant and trace of the matrix.

For the 3x3 matrix A:

                   A = `[[A_11,A_12, A_13],[A_21,A_22,A_23],[A_31,A_32,A_33]]`,

the characteristic polynomial can be found using the formula:

   CP =  -λ³+ tr(A)λ² - 1/2( tr(A)² - tr(A²)) λ + det(A),

where:

• tr(A) is the trace of 3x3 matrix
• det(A) is the determinant of 3x3 matrix

Characteristic Polynomial for a 2x2 Matrix

For the Characteristic Polynomial of a 2x2 matrix, CLICK HERE