Eigenvector of a 3x3 Matrix Calculator

This Eigenvector Calculator finds the eigenvectors of a 3×3 matrix. It first finds the eigenvalues, then for each eigenvalue solves (A − λI)v = 0 to get the matching eigenvector.

Step-by-step method

1. Write the 3x3 matrix A.
2. Write the eigenvector equation (A - λ I)v = 0.
3. Find the eigenvalues first.
4. For each eigenvalue, solve for the eigenvector.

Formula:

Example 1:

Step 1 - Write the 3x3 matrix A.

In this problem: We find the eigenvectors of the matrix \(A = \begin{pmatrix} 2 & 0 & 0 \\ 1 & 2 & 1 \\ -1 & 0 & 1 \end{pmatrix}\).

Step 2 - Write the eigenvector equation (A - λ I)v = 0.

In this problem: For each eigenvalue, eigenvectors solve (A - \(\lambda\) I)v = 0.

Step 3 - Find the eigenvalues first.

In this problem: The eigenvalues are \(\lambda = 1,\; \lambda = 2,\; \lambda = 2\).

Step 4a - For each eigenvalue, solve for the eigenvector.

In this problem: For \(\lambda = 1\), solve \((A - \lambda I)v = 0\) to get the eigenvector.

Step 4b - For each eigenvalue, solve for the eigenvector.

In this problem: For \(\lambda = 2\), solve \((A - \lambda I)v = 0\) to get the eigenvector.

Step 4c - For each eigenvalue, solve for the eigenvector.

In this problem: For \(\lambda = 2\), solve \((A - \lambda I)v = 0\) to get the eigenvector.

Final answer:

Example 2:

Step 1 - Write the 3x3 matrix A.

In this problem: We find the eigenvectors of the matrix \(A = \begin{pmatrix} 2 & -3 & 1 \\ 1 & -2 & 1 \\ 1 & -3 & 2 \end{pmatrix}\).

Step 2 - Write the eigenvector equation (A - λ I)v = 0.

In this problem: For each eigenvalue, eigenvectors solve (A - \(\lambda\) I)v = 0.

Step 3 - Find the eigenvalues first.

In this problem: The eigenvalues are \(\lambda = 0,\; \lambda = 1,\; \lambda = 1\).

Step 4a - For each eigenvalue, solve for the eigenvector.

In this problem: For \(\lambda = 0\), solve \((A - \lambda I)v = 0\) to get the eigenvector.

Step 4b - For each eigenvalue, solve for the eigenvector.

In this problem: For \(\lambda = 1\), solve \((A - \lambda I)v = 0\) to get the eigenvector.

Step 4c - For each eigenvalue, solve for the eigenvector.

In this problem: For \(\lambda = 1\), solve \((A - \lambda I)v = 0\) to get the eigenvector.

Final answer:

See Example 1 Hide Example 1

Problem

\(A = \begin{pmatrix} 2 & 0 & 0 \\ 1 & 2 & 1 \\ -1 & 0 & 1 \end{pmatrix}\)

Approach

Step-by-step method

1. Write the 3x3 matrix A.
2. Write the eigenvector equation (A - λ I)v = 0.
3. Find the eigenvalues first.
4. For each eigenvalue, solve for the eigenvector.

Step 1

Step 1 - Write the 3x3 matrix A.

In this problem: We find the eigenvectors of the matrix \(A = \begin{pmatrix} 2 & 0 & 0 \\ 1 & 2 & 1 \\ -1 & 0 & 1 \end{pmatrix}\).

\(A = \begin{pmatrix} 2 & 0 & 0 \\ 1 & 2 & 1 \\ -1 & 0 & 1 \end{pmatrix}\)

Step 2

Step 2 - Write the eigenvector equation (A - λ I)v = 0.

In this problem: For each eigenvalue, eigenvectors solve (A - \(\lambda\) I)v = 0.

\((A - \lambda I)\, v = 0\)

Step 3

Step 3 - Find the eigenvalues first.

In this problem: The eigenvalues are \(\lambda = 1,\; \lambda = 2,\; \lambda = 2\).

\(\lambda = 1,\; \lambda = 2,\; \lambda = 2\)

Step 4a

Step 4a - For each eigenvalue, solve for the eigenvector.

In this problem: For \(\lambda = 1\), solve \((A - \lambda I)v = 0\) to get the eigenvector.

\((A - 1 I) = \begin{pmatrix} 1 & 0 & 0 \\ 1 & 1 & 1 \\ -1 & 0 & 0 \end{pmatrix} \;\Rightarrow\; v = \begin{pmatrix} 0 \\ -1 \\ 1 \end{pmatrix}\)

Step 4b

Step 4b - For each eigenvalue, solve for the eigenvector.

In this problem: For \(\lambda = 2\), solve \((A - \lambda I)v = 0\) to get the eigenvector.

\((A - 2 I) = \begin{pmatrix} 0 & 0 & 0 \\ 1 & 0 & 1 \\ -1 & 0 & -1 \end{pmatrix} \;\Rightarrow\; v = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}\)

Step 4c

Step 4c - For each eigenvalue, solve for the eigenvector.

In this problem: For \(\lambda = 2\), solve \((A - \lambda I)v = 0\) to get the eigenvector.

\((A - 2 I) = \begin{pmatrix} 0 & 0 & 0 \\ 1 & 0 & 1 \\ -1 & 0 & -1 \end{pmatrix} \;\Rightarrow\; v = \begin{pmatrix} -1 \\ 0 \\ 1 \end{pmatrix}\)

Final Answer

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See Example 2 Hide Example 2

Problem

\(A = \begin{pmatrix} 2 & -3 & 1 \\ 1 & -2 & 1 \\ 1 & -3 & 2 \end{pmatrix}\)

Approach

Step-by-step method

1. Write the 3x3 matrix A.
2. Write the eigenvector equation (A - λ I)v = 0.
3. Find the eigenvalues first.
4. For each eigenvalue, solve for the eigenvector.

Step 1

Step 1 - Write the 3x3 matrix A.

In this problem: We find the eigenvectors of the matrix \(A = \begin{pmatrix} 2 & -3 & 1 \\ 1 & -2 & 1 \\ 1 & -3 & 2 \end{pmatrix}\).

\(A = \begin{pmatrix} 2 & -3 & 1 \\ 1 & -2 & 1 \\ 1 & -3 & 2 \end{pmatrix}\)

Step 2

Step 2 - Write the eigenvector equation (A - λ I)v = 0.

In this problem: For each eigenvalue, eigenvectors solve (A - \(\lambda\) I)v = 0.

\((A - \lambda I)\, v = 0\)

Step 3

Step 3 - Find the eigenvalues first.

In this problem: The eigenvalues are \(\lambda = 0,\; \lambda = 1,\; \lambda = 1\).

\(\lambda = 0,\; \lambda = 1,\; \lambda = 1\)

Step 4a

Step 4a - For each eigenvalue, solve for the eigenvector.

In this problem: For \(\lambda = 0\), solve \((A - \lambda I)v = 0\) to get the eigenvector.

\((A - 0 I) = \begin{pmatrix} 2 & -3 & 1 \\ 1 & -2 & 1 \\ 1 & -3 & 2 \end{pmatrix} \;\Rightarrow\; v = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}\)

Step 4b

Step 4b - For each eigenvalue, solve for the eigenvector.

In this problem: For \(\lambda = 1\), solve \((A - \lambda I)v = 0\) to get the eigenvector.

\((A - 1 I) = \begin{pmatrix} 1 & -3 & 1 \\ 1 & -3 & 1 \\ 1 & -3 & 1 \end{pmatrix} \;\Rightarrow\; v = \begin{pmatrix} 3 \\ 1 \\ 0 \end{pmatrix}\)

Step 4c

Step 4c - For each eigenvalue, solve for the eigenvector.

In this problem: For \(\lambda = 1\), solve \((A - \lambda I)v = 0\) to get the eigenvector.

\((A - 1 I) = \begin{pmatrix} 1 & -3 & 1 \\ 1 & -3 & 1 \\ 1 & -3 & 1 \end{pmatrix} \;\Rightarrow\; v = \begin{pmatrix} -1 \\ 0 \\ 1 \end{pmatrix}\)

Final Answer

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