1.4.12 Matrix transformations

Condition: For each eigenvalue \( \lambda \) of the matrix \( A=\left(\begin{array}{ccc}41 / 25 & 0 & 12 / 25 \\ 0 & -3 & 0 \\ 12 / 25 & 0 & 34 / 25\end{array}\right) \) using the Gaussian method, find all its eigenvectors \( \bar{x} \neq \overline{0} \) and check all found pairs \( \{\lambda, \bar{x}\} \) (should be \( A \bar{x}=\lambda \bar{x} \) ).