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Problem 242

# Find the eigenvalues and the eigenvectors of the matrix $$\mathrm{A}$$, where

Expert verified
The eigenvalues λ₁ and λ₂ are the solutions of the quadratic equation $$λ^2 - λ(a+d) + (ad - bc) = 0$$. For each eigenvalue, find the corresponding eigenvectors by solving the system of linear equations $$(A - λI)X = 0$$, where A is the given matrix and I is the identity matrix. The eigenvectors can be obtained using row-reduction or any other preferred method.
See the step by step solution

## Step 1: Compute the determinant of (A - λI)

Given the matrix A, we first need to compute the determinant of the matrix (A - λI), where λ is the eigenvalue and I is the identity matrix. Let A = $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ To obtain (A- λI), we subtract λ from the diagonal elements: $\begin{bmatrix} a-λ & b \\ c & d-λ \end{bmatrix}$ Now, find the determinant |A - λI|: $$|A - λI| = (a - λ)(d - λ) - bc$$

## Step 2: Solve for λ

Now we need to set the determinant equation equal to zero and solve for λ: $$(a - λ)(d - λ) - bc = 0$$ By expanding the equation, we get a quadratic equation in terms of λ: $$λ^2 - λ(a+d) + (ad - bc) = 0$$ Solve the quadratic equation to obtain the eigenvalues λ₁ and λ₂.

## Step 3: Find the eigenvectors

Now, for each eigenvalue, λ, we need to find the eigenvectors that satisfy the equation ($$A - λI)X = 0$$, where X is the eigenvector. For λ₁: 1. Substitute λ₁ into the matrix (A-λI): $\begin{bmatrix} a-λ₁ & b \\ c & d-λ₁ \end{bmatrix}$ 2. Solve the system of linear equations ($$A-λ₁I)X = 0$$ either by row-reduction or any other preferred method: - If only one equation is left, then we have a one-parameter solution (or more), and we can parameterize the eigenvector X. - If two independent equations are left, then there are no eigenvectors for this eigenvalue. Repeat the same process for λ₂. Now, you have found the eigenvalues (λ₁, λ₂) and their corresponding eigenvectors.

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