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}}