## Introduction to a Characteristic Polynomial Calculator:

Characteristic polynomial calculator is an online tool that is designed to **determine the characteristic polynomial** of a given square matrix. Our calculator simplifies the process of finding eigenvalues from the characteristic polynomial equation.

It's a valuable tool for students, researchers, and professionals, who are working with matrices and linear algebra to get solutions without manual calculation.

## What is a Characteristic Polynomial?

Characteristic Polynomial of a square matrix A is defined as the **determinant equation** where an indeterminate (variable) λ commutes with the identity matrix of the same order as A has to find the eigenvalues. It is denoted as p(λ).

The characteristic polynomial roots are the eigenvalues of the square matrix A. The equation can be written as follows:

$$ p(λ) \;=\; det(A−λI) $$

## Calculation Process in Characteristic Equation Calculator:

Characteristic polynomial of a matrix calculator provides a straightforward method to calculate the characteristic polynomial of any square matrix, for eigenvalues. Follow these simple steps such as:

**Step 1:** Identify the given square matrix A in the **dimensions n×n** order. The elements of a given matrix can be real numbers or complex numbers depending on the type of matrix.

**Step 2:** To make the given matrix in the form of a characteristic polynomial equation such as IA−λII, subtract matrix A with an indeterminate λ and I matrix.

**Step 3:** Calculate the determinant of the characteristic polynomial equation to find the eigenvalue such as:

$$ p(λ) \;=\; det\; (A−λI) $$

**Step 4:** The characteristic equation solver gives the characteristic polynomial p(λ) in a simplified expression. This polynomial provides valuable information about the eigenvalues of the original matrix A.

## Solved Example of the Characteristic Polynomial

Let's see an **example** of a characteristic polynomial equation with a solution to get better clarity about the working process of the characteristic equation of matrix calculator.

### Example: Find Matrix Characteristics Polynomial,

$$ \left[ \begin{matrix} 5 & -6 & 2 \\ -3 & 1 & -4 \\ 2 & -4 & 3 \\ \end{matrix} \right] $$

**Solution**:

$$ A - \lambda I $$

$$ =\; \left| \begin{matrix} (5 - \lambda ) & -6 & 2 \\ -3 & (1 - \lambda ) & -4 \\ 2 & -4 & (3 - \lambda ) \\ \end{matrix} \right| $$

$$ =\; (5 - \lambda)((1 - \lambda) \times (3 - \lambda) - (-4) \times (-4)) - (-6)((-3) \times (3 - \lambda) - (-4) \times 2) + 2((-3) \times (-4) - (1 - \lambda) \times 2 $$

$$ =\; (5 - \lambda) \biggr( \biggr(3 - 4\lambda + \lambda^2 \biggr) - 16 \biggr) + 6 ((-9 + 3 \lambda) - (-8)) + 2( 12 - (2 - 2 \lambda)) $$

$$ =\; (5 - \lambda) \biggr( -13 - 4\lambda + \lambda^2 \biggr) + 6(-1 + 3 \lambda) + 2(10 + 2 \lambda) $$

$$ =\; \biggr( -65 - 7\lambda + 9 \lambda^2 - \lambda^3 \biggr) + (-6 + 18 \lambda) + (20 + 4 \lambda) $$

$$ =\; \biggr(- \lambda^3 + 9 \lambda^2 + 15 \lambda - 51 \biggr) $$

$$ =\; - \biggr(\lambda^3 - 9 \lambda^2 - 15 \lambda + 51 \biggr) $$

## How to Use the Characteristic Polynomial Calculator?

Characteristic equation calculator has a simple design that makes it easy for you to know how to use it for the evaluation of characteristic polynomial equations, only when you follow our instructions that are given as:

- Choose the size of the matrix from the given field for the characteristic equation.
- Enter the element of a square matrix to find the characteristic equation in the input field.
- Review the given matrix before hitting the calculate button to start the evaluation process in the characteristic equation calculator.
- Click the “
**Calculate**” button to get the result of your given characteristic polynomial problem. - If you are trying our characteristic polynomial of a matrix calculator for the first time then you can use the load example to learn more about this method.
- Click on the “Recalculate” button to get a new page for finding more example solutions of characteristic polynomial problems to get eigenvalue.

## Final Result of Characteristic Equation Solver:

Characteristic equation of matrix calculator gives you the **solution** from a given characteristic polynomial when you add the input into it. It included as:

**Result Option**

When you click on the result option the characteristic polynomial calculator gives you a solution to the square matrix problem to find characteristic polynomials.

**Possible Steps**

When you click on it, this option will provide you with a solution where all the calculations of characteristic polynomial process steps are mentioned.

## Benefits of Using the Characteristic Equation Calculator:

The characteristic polynomial of a matrix calculator provides you with many benefits that help you to calculate square matrix problems and give you solutions without any trouble. These benefits are:

- A characteristic equation solver is a
**free-of-cost tool**so you can use it for free to find characteristic polynomial problem solutions without paying anything. - It is an adaptable tool that can manage various types of matrices to calculate the characteristic polynomial equation.
- Our characteristic equation of matrix calculator helps you to get conceptual clarity for the characteristic polynomial process when you use it for practice by solving more examples.
- It saves the time that you consume on the calculation of the characteristic polynomial problems.
- Characteristic polynomial calculator is a reliable tool that provides you with accurate solutions whenever you use it to calculate the characteristic polynomial without any man-made error in evaluation.
- A characteristic equation calculator provides the solution without imposing any restriction which means you can use it multiple times.