# Characteristic Polynomial of a 3x3 Matrix

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SavannahBergen.Characteristic Polynomial of a 3x3 Matrix
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1fe0a0b6-1ea2-11e6-9770-bc764e2038f2

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

INSTRUCTIONS: Enter the following:

• (A)  This is the 3x3 matrix.

Polynomial: The calculator returns the polynomial.

Related Calculators:

• To compute the product of a 3x3 matrix and a 3x1 matrix, CLICK HERE.

### 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 -λ^3+"tr"(A)λ^2+("tr"(A)^2-"tr"(A^2))λ+det(A), where "tr"(A) is the trace of A and det(A) is the determinant of A.

### Characteristic Polynomial for a 2x2 Matrix

This equation, Characteristic Polynomial of a 3x3 Matrix, is listed in 1 Collection.