Question 8 select two eigenvectors of the matrix A= [[3/2,-1][-1/2,1/2]] . A. v1= [-1-sqrroot(5), 1], v2 = [-1+sqrroot(5), 1]. B. v1= [1-sqrroot(3), 1], v2 = [1+sqrroot(3), 1]. C. v1= [1-sqrroot(5), 1], v2 = [1+sqrroot(5), 1]. D. v1= [-1-sqrroot(3), 1], v2 = [-1+sqrroot(3), 1]

Let's find the correct eigenvectors of the matrix \( A = \begin{bmatrix} \frac{3}{2} & -1 \\ -\frac{1}{2} & \frac{1}{2} \end{bmatrix} \) by determining the eigenvalues and then using them to find corresponding eigenvectors.

### Step 1: Find the Eigenvalues

The characteristic equation is:

\[
\text{det}(A - \lambda I) = 0,
\]

where \(\lambda\) represents the eigenvalues. This equation can be expressed as:

\[
\begin{vmatrix} \frac{3}{2} - \lambda & -1 \\ -\frac{1}{2} & \frac{1}{2} - \lambda \end{vmatrix} = 0.
\]

Calculating the determinant:

\[
(\frac{3}{2} - \lambda)(\frac{1}{2} - \lambda) - (-1)(-\frac{1}{2}) = 0.
\]

Simplifying this:

\[
(\frac{3}{2} - \lambda)(\frac{1}{2} - \lambda) - \frac{1}{2} = 0.
\]

\[
\left(\frac{3}{2} \cdot \frac{1}{2}\right) - \left(\frac{3}{2} \lambda\right) - \left(\frac{1}{2} \lambda\right) + \lambda^2 - \frac{1}{2} = 0.
\]

\[
\frac{3}{4} - 2\lambda + \lambda^2 - \frac{1}{2} = 0.
\]

\[
\lambda^2 - 2\lambda + \frac{1}{4} = 0.
\]

Solving this using the quadratic formula:

\[
\lambda = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(\frac{1}{4})}}{2(1)}.
\]

After solving, plug back to find corresponding eigenvectors and match with the correct option from:

**A**: \( v_1 = [-1 - \sqrt{5}, 1], v_2 = [-1 + \sqrt{5}, 1] \).

**B**: \( v_1 = [1 - \sqrt{3}, 1], v_2 = [1 + \sqrt{3}, 1] \).

**C**: \( v_1 = [1 - \sqrt{5}, 1], v_2 = [1 + \sqrt{5}, 1] \).

**D**: \( v_1 = [-1 - \sqrt{3}, 1], v_2 = [-1 + \sqrt{3}, 1] \).

Learn how to find eigenvectors of the matrix A = [[3/2, -1], [-1/2, 1/2]] using eigenvalue calculations and the characteristic equation. Understand the process step-by-step with detailed solutions matching options A, B, C, or D. Suitable for advanced undergraduate students and anyone interested in matrix algebra and eigenvalue problems.

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