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# Eigenvalue Calculator

e Calculator helps to calculate the eigenvalues of a 2 × 2 matrix. Eigenvalues are associated with eigenvectors and are used to analyze linear transformations.

## What is Eigenvalue Calculator?

Eigenvalue Calculator is an online tool that helps to compute the eigenvalues for a given 2 × 2 matrix. Eigenvalues can also be defined as a special set of scalars that are associated with a system of linear equations. To use the * eigenvalue calculator*, enter the values in the given input boxes.

### Eigenvalue Calculator

## How to Use Eigenvalue Calculator?

Please follow the steps below to compute the eigenvalues of a 2 × 2 matrix using the eigenvalue calculator.

**Step 1:**Go to Cuemath's online eigenvalue calculator.**Step 2:**Enter the values in the given input boxes of the eigenvalue calculator.**Step 3:**Click on the**"Calculate"**button to find the eigenvalues for a given 2 x 2 matrix.**Step 4:**Click on the**"Reset"**button to clear the fields and enter new values.

## How Does Eigenvalue Calculator Work?

Suppose we have a square matrix given by \(A_{n\times n}\). Let λ be a scalar quantity. Then [A - λI] is known as an Eigen or a characteristic matrix. Here, I is used to represent the identity matrix. The determinant of this characteristic matrix can be given by | A - λI | and the Eigen equation will be | A - λI | = 0. To find the Eigenvalues of this equation the following steps are utilized.

- Let A be a 2 × 2 square matrix.
- The identity matrix I = \(\begin{bmatrix} 1 & 0\\ 0& 1 \\\end{bmatrix}\)
- Now we multiply the identity matrix I with some scalar λ. This gives us λI
- Next, we subtract λI from the matrix A; A - λI
- We then find the determinant of the matrix obtained. That is | A - λI |.
- This results in a quadratic expression.
- We equate this expression to zero. Thus, | A - λI | = 0.
- Finally, we solve the quadratic equation to get two values of λ. These two values will be the eigenvalues.

## Solved Examples on Eigenvalue Calculator

**Example 1:**

Find the eigenvalue of \(\begin{bmatrix} 0 & 1\\ 2& 3 \\\end{bmatrix}\) and verify it using the eigenvalue calculator.

**Solution:**

Given matrix: A = \(\begin{bmatrix} 0 & 1\\ 2& 3 \\\end{bmatrix}\)

| A - λI | = 0, where I is identity matrix i.e.,

A - λI = \(\begin{bmatrix} 0 & 1\\ 2& 3 \\\end{bmatrix}\) - λ\(\begin{bmatrix} 1 & 0\\ 0& 1 \\\end{bmatrix}\)

A - λI = \(\begin{bmatrix} - λ & 1\\ 2 & 3 - λ \\\end{bmatrix}\)

| A - λI | = λ^{2} - 3λ - 2

Substituting these values in | A - λI | = 0 we get

λ^{2} - 3λ - 2 = 0

On solving,

λ = -0.56 , 3.56

**Example 2:**

Find the eigenvalue of \(\begin{bmatrix} 1.2 & 3.4\\ 1.6& 3 \\\end{bmatrix}\) and verify it using the eigenvalue calculator.

**Solution:**

Given matrix: A = \(\begin{bmatrix} 1.2 & 3.4\\ 1.6& 3 \\\end{bmatrix}\)

| A - λI | = 0, where I is identity matrix i.e.,

A - λI = \(\begin{bmatrix} 1.2 & 3.4\\ 1.6& 3 \\\end{bmatrix}\) - λ\(\begin{bmatrix} 1 & 0\\ 0& 1 \\\end{bmatrix}\)

A - λI = \(\begin{bmatrix} 1.2 - λ & 3.4\\ 1.6 & 3 - λ \\\end{bmatrix}\)

| A - λI | = λ^{2} - 4.2λ - 1.84

Substituting these values in | A - λI | = 0 we get

λ^{2} - 4.2λ - 1.84 = 0

On solving,

λ = -0.4 , 4.6

Similarly, you can try the eigenvalue calculator to find the value of the eigenvalues for the following matrices:

- \(\begin{bmatrix} 8 & 9\\ 10& 5.6 \\\end{bmatrix}\)
- \(\begin{bmatrix} 0.25 & 1.3\\ 7.8& 0.5 \\\end{bmatrix}\)

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