Eigenvector of a 2x2 Matrix Calculator

Published on: July 18, 2026
Final Answer: Free • Full Steps: Pro

This Eigenvector Calculator finds the eigenvectors of a 2×2 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 2x2 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:

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

Example 1:

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

Step 1 - Write the 2x2 matrix A.

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

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

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

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

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

Step 3 - Find the eigenvalues first.

In this problem: The eigenvalues are \(\lambda = 1,\; \lambda = 3\).

\(\lambda = 1,\; \lambda = 3\)

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.

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

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

In this problem: For \(\lambda = 3\), solve \((A - \lambda I)v = 0\) to get the eigenvector.

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

Final answer:

\\(\lambda = 1:\; v = \begin{pmatrix} -1 \\ 1 \end{pmatrix},\quad \lambda = 3:\; v = \begin{pmatrix} 1 \\ 1 \end{pmatrix}\\)

Example 2:

\\(A = \begin{pmatrix} 4 & -2 \\ 1 & 1 \end{pmatrix}\\)

Step 1 - Write the 2x2 matrix A.

In this problem: We find the eigenvectors of the matrix \(A = \begin{pmatrix} 4 & -2 \\ 1 & 1 \end{pmatrix}\).

\(A = \begin{pmatrix} 4 & -2 \\ 1 & 1 \end{pmatrix}\)

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

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

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

Step 3 - Find the eigenvalues first.

In this problem: The eigenvalues are \(\lambda = 2,\; \lambda = 3\).

\(\lambda = 2,\; \lambda = 3\)

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

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

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

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

In this problem: For \(\lambda = 3\), solve \((A - \lambda I)v = 0\) to get the eigenvector.

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

Final answer:

\\(\lambda = 2:\; v = \begin{pmatrix} 1 \\ 1 \end{pmatrix},\quad \lambda = 3:\; v = \begin{pmatrix} 2 \\ 1 \end{pmatrix}\\)
See Example 1 Hide Example 1

Problem

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

Approach

Step-by-step method

  1. Write the 2x2 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 2x2 matrix A.

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

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

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 \(\lambda = 1,\; \lambda = 3\).

\(\lambda = 1,\; \lambda = 3\)

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.

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

Step 4b

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

In this problem: For \(\lambda = 3\), solve \((A - \lambda I)v = 0\) to get the eigenvector.

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

Final Answer

\(\)
See Example 2 Hide Example 2

Problem

\(A = \begin{pmatrix} 4 & -2 \\ 1 & 1 \end{pmatrix}\)

Approach

Step-by-step method

  1. Write the 2x2 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 2x2 matrix A.

In this problem: We find the eigenvectors of the matrix \(A = \begin{pmatrix} 4 & -2 \\ 1 & 1 \end{pmatrix}\).

\(A = \begin{pmatrix} 4 & -2 \\ 1 & 1 \end{pmatrix}\)

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 \(\lambda = 2,\; \lambda = 3\).

\(\lambda = 2,\; \lambda = 3\)

Step 4a

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

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

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

Step 4b

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

In this problem: For \(\lambda = 3\), solve \((A - \lambda I)v = 0\) to get the eigenvector.

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

Final Answer

\(\)
Matrix A