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# Characteristic Polynomial of a 3x3 Matrix

Variable | Instructions | Datatype |
---|---|---|

`(A)" 3x3 Matrix"` | Enter the 3x3 matrix. | Matrix |

SavannahBergen.Characteristic Polynomial of a 3x3 Matrix

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.
- To compute the Mirror of a 3x3 Matrix, CLICK HERE.
- To compute the Inverse of a 3x3 Matrix, CLICK HERE.
- To compute the Transpose of a 3x3 Matrix, CLICK HERE.
- To compute the Trace of a 3x3 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

For theCharacteristic Polynomial of a 2x2 matrix, CLICK HERE

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