Is [ 1   (7+√85) /6 ] is eigen vector of [ 1/3  1/2   1/2  3/2] ( 2 × 2 matrix)

To check whether the given vector is an eigenvector of the matrix, we need to multiply the matrix by the vector and verify if the result is a scalar multiple of the vector. This scalar would be the corresponding eigenvalue.

### Given:
- Matrix \( A = \begin{pmatrix} \frac{1}{3} & \frac{1}{2} \\ \frac{1}{2} & \frac{3}{2} \end{pmatrix} \)
- Vector \( v = \begin{pmatrix} 1 \\ \frac{7 + \sqrt{85}}{6} \end{pmatrix} \)

We perform matrix-vector multiplication \( A \cdot v \) and check if the result is a scalar multiple of \( v \).

Let's compute the product:

\[
A \cdot v = \begin{pmatrix} \frac{1}{3} & \frac{1}{2} \\ \frac{1}{2} & \frac{3}{2} \end{pmatrix} \cdot \begin{pmatrix} 1 \\ \frac{7 + \sqrt{85}}{6} \end{pmatrix}
\]

I will calculate this step by step.The result of multiplying the matrix \( A \) by the vector \( v \) is:

\[
A \cdot v = \begin{pmatrix} 1.68496204 \\ 4.55488611 \end{pmatrix}
\]

Now, we check if this result is a scalar multiple of the vector \( v = \begin{pmatrix} 1 \\ \frac{7 + \sqrt{85}}{6} \end{pmatrix} \approx \begin{pmatrix} 1 \\ 2.6899441 \end{pmatrix} \).

We divide the components of the result by the corresponding components of the vector \( v \) to check if the same scalar appears for both components.

Let me check that.The ratio of the components of \( A \cdot v \) to the components of \( v \) is approximately \( 1.68496204 \) for both components. This confirms that \( A \cdot v \) is a scalar multiple of \( v \), and the scalar multiple (eigenvalue) is approximately \( 1.685 \).

Thus, the vector \( v = \begin{pmatrix} 1 \\ \frac{7 + \sqrt{85}}{6} \end{pmatrix} \) is indeed an eigenvector of the matrix \( A \), with an eigenvalue of approximately \( 1.685 \).

Would you like more details on eigenvectors, or any questions about this process?

Here are five related questions:
1. How do you determine the eigenvalue corresponding to an eigenvector?
2. Can a matrix have more than one eigenvector?
3. What is the significance of eigenvectors and eigenvalues in linear transformations?
4. How can you compute the eigenvalues of a matrix directly?
5. What types of matrices have real eigenvalues?

**Tip:** Eigenvectors and eigenvalues often reveal important geometric properties of a matrix, such as directions of stretching or compression in transformations.

Is [ 1   (7+√85) /6 ] an eigenvector of [ 1/3  1/2   1/2  3/2] (2 × 2 matrix)?

Verify if the vector [1, (7 + √85) / 6] is an eigenvector of the matrix [1/3, 1/2; 1/2, 3/2]. This problem covers essential linear algebra concepts such as eigenvectors, eigenvalues, and matrix multiplication. A step-by-step solution shows how to perform matrix-vector multiplication and check for scalar multiples.

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