# Characteristic Polynomial Calculator

## Find the characteristic polynomial of a matrix step by step

The calculator will find the characteristic polynomial of the given matrix, with steps shown.

The Characteristic Polynomial Calculator is our advanced tool that allows you to compute the characteristic polynomial of any square matrix efficiently, significantly reducing the time and effort required for manual calculations.

The characteristic polynomial is a polynomial equation derived from a square matrix that holds crucial information about the matrix's properties and behavior.

## How to Use the Characteristic Polynomial Calculator?

### Input Square Matrix

Enter the elements of your square matrix into the designated input field. You can typically input the matrix as rows of numbers, separating the elements with commas or spaces. Use semicolons or new lines to separate rows.

### Initiate Calculation

Once you have entered the square matrix, click the "Calculate" button to initiate the computation process. There is no need for additional settings or parameters.

### View the Result

The calculator will instantly display the characteristic polynomial of the entered square matrix. The characteristic polynomial is typically shown in the form of an equation with the variable $$$\lambda$$$ representing the eigenvalues.

## What Is a Characteristic Polynomial?

The characteristic polynomial $$$p(\lambda)$$$ of a square matrix $$$A$$$ is a polynomial equation that helps determine the matrix's eigenvalues, explore its properties, and solve systems of linear equations. It is obtained by subtracting the scalar $$$\lambda$$$ times the identity matrix $$$I$$$ from the given square matrix $$$A$$$, and then calculating the determinant (det) of the resulting matrix. The resulting polynomial equation, $$$p(\lambda)=\operatorname{det}(A-\lambda I)$$$, is referred to as the characteristic polynomial.

For example, consider a 2x2 square matrix $$$A$$$:

$$A=\left[\begin{array}{cc}a&b\\c&d\end{array}\right]$$To find the characteristic polynomial $$$p(\lambda)$$$ for this matrix, we subtract $$$\lambda$$$ times the identity matrix:

$$A-\lambda I=\left[\begin{array}{cc}a-\lambda&b\\c&d-\lambda\end{array}\right]$$Next, we calculate the determinant of $$$A-\lambda I$$$:

$$\operatorname{det}(A-\lambda I)=(a-\lambda)(d-\lambda)-bc$$The resulting polynomial equation in $$$\lambda$$$ becomes the characteristic polynomial $$$p(\lambda)$$$ for the matrix $$$A$$$:

$$p(\lambda)=(a-\lambda)(d-\lambda)-bc$$This characteristic polynomial helps identify the eigenvalues of the matrix $$$A$$$, study its properties, and solve various linear algebraic problems related to $$$A$$$.

## Determining the Characteristic Polynomial of a 3x3 Matrix

Determining the characteristic polynomial of a 3x3 matrix is a crucial step in understanding its properties and behavior. By computing the characteristic polynomial, you can obtain valuable information about its eigenvalues, which play a significant role in various mathematical applications.

To determine the characteristic polynomial of a 3x3 matrix, follow these steps:

Begin with a given 3x3 matrix, denoted as $$$A$$$:

$$A=\left[\begin{array}{ccc}a&b&c\\d&e&f\\g&h&i\end{array}\right]$$Subtract $$$\lambda$$$ times the identity matrix $$$I$$$ from the matrix $$$A$$$, where $$$\lambda$$$ is a scalar variable representing the eigenvalues:

$$A-\lambda I=\left[\begin{array}{ccc}a-\lambda&b&c\\d&e-\lambda&f\\g&h&i-\lambda\end{array}\right]$$Calculate the determinant (det) of $$$A-\lambda I$$$:

$$\operatorname{det}(A-\lambda I)=(a-\lambda)((e-\lambda)(i-\lambda)-fh)-b((d-\lambda)(i-\lambda)-fg)+c((d-\lambda)h-eg)$$Simplify the expression and expand the determinant:

$$\operatorname{det}(A-\lambda I)=-\lambda^3+(a+e+i)\lambda^2-(ae+ai+ei-fh-dg-ch)\lambda+(aei+bfg+cdh-ceg-afh -bdi)$$The resulting polynomial equation in $$$\lambda$$$ is the characteristic polynomial $$$p(\lambda)$$$:

$$p(\lambda)=-\lambda^3+(a+e+i)\lambda^2-(ae+ai+ei-fh-dg-ch)\lambda+(aei+bfg+cdh-ceg-afh -bdi)$$Understanding the characteristic polynomial of a 3x3 matrix is pivotal for exploring its eigenvalues and properties. This polynomial acts as a cornerstone for conducting advanced analysis and computations in linear algebra. By accurately determining the characteristic polynomial, you delve into the underlying mathematical characteristics of the matrix, allowing you to make informed decisions in a variety of disciplines and applications.

## Why Choose Our Characteristic Polynomial Calculator?

### Efficiency

Our calculator swiftly computes the characteristic polynomial, saving you valuable time and effort compared to manual calculations. It automates complex computations, providing results in an instant.

### Accuracy

Built on reliable mathematical algorithms and formulas, our calculator ensures accurate and precise results. You can rely on it for dependable calculations, eliminating the possibility of human errors.

### Comprehensive Solutions

Our calculator caters specifically to 3x3 matrices, providing tailored solutions to meet your needs. It enables you to gain a deep understanding of the matrix's eigenvalues and properties, empowering you to make informed decisions in your mathematical explorations.

### Educational Resource

In addition to calculating the characteristic polynomial, our calculator serves as an educational resource. It offers an opportunity to learn and comprehend the concept of characteristic polynomials, expanding your knowledge of linear algebra.

### FAQ

#### Can I use the results generated by the Characteristic Polynomial Calculator for further analysis?

Absolutely! You can use the characteristic polynomial obtained from our calculator for further analysis, such as identifying eigenvalues, exploring eigenvectors, and studying the matrix's properties.

#### What other information can I obtain from the characteristic polynomial?

The characteristic polynomial provides insight into the matrix's eigenvalues, which play a crucial role in understanding its behavior, diagonalizability, and solving systems of linear equations.

#### Is the Characteristic Polynomial Calculator accurate?

Yes, our calculator uses correct mathematical formulas and reliable algorithms to ensure accurate calculations. However, it's always good practice to double-check your inputs and results.

#### What should I do if the calculator isn't working?

First, make sure your inputs are correct. If the issue persists, refresh the page or contact our support team.