2 السوig
Find the Eigenvalues and Eigenvectors of the following matrix
\[
\left|\begin{array}{cc}
-0 & r \\
r & -r
\end{array}\right|
\]
\[
\begin{array}{l}
(-1,7),\left(\begin{array}{l}
1 \\
r
\end{array}\right),\left(\begin{array}{c}
-r \\
1
\end{array}\right) 0 \\
(-1,-7),\left(\begin{array}{l}
1 \\
r
\end{array}\right),\left(\begin{array}{c}
r \\
-1
\end{array}\right) 0 \\
(1,7),\left(\begin{array}{l}
1 \\
r
\end{array}\right),\left(\begin{array}{c}
r \\
-1
\end{array}\right) 0 \\
(1,-7),\left(\begin{array}{c}
-1 \\
r
\end{array}\right),\left(\begin{array}{c}
r \\
-1
\end{array}\right) 0
\end{array}
\]
فنظ الإجابة 4

Step 1 ：Find the eigenvalues by solving the characteristic equation: \(\det(A - \lambda I) = 0\)

Step 2 ：Eigenvalues: \(\lambda_1 = 1\), \(\lambda_2 = -1\)

Step 3 ：Find the eigenvectors by plugging the eigenvalues back into the matrix equation: \((A - \lambda I)\vec{v} = 0\)

Step 4 ：Eigenvectors: \(\vec{v}_1 = \begin{bmatrix} -\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} \end{bmatrix}\), \(\vec{v}_2 = \begin{bmatrix} -\frac{1}{\sqrt{2}} \\ -\frac{1}{\sqrt{2}} \end{bmatrix}\)

Step 5 ：\boxed{\text{Eigenvalues: } \lambda_1 = 1, \lambda_2 = -1 \text{ and Eigenvectors: } \vec{v}_1 = \begin{bmatrix} -\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} \end{bmatrix}, \vec{v}_2 = \begin{bmatrix} -\frac{1}{\sqrt{2}} \\ -\frac{1}{\sqrt{2}} \end{bmatrix}}