1.4.11 Matrix transformations

( underline{ ext { Condition: }} ) Find all eigenvalues ​​of the matrix ( A=left( egin{array}{ccc}41 / 25 & 0 & 12 / 25 \ 0 & -3 & 0 \ 12 / 25 & 0 & 34 / 25end{array} ight) ), check them out. After obtaining the characteristic equation, check its free term (should be ( operatorname{det} A ) ) and its coefficient at ( lambda^{2} ) (should be ( operatorname{tr} A ) ). When solving this equation, select the first root ( lambda_{1}), choosing it among the divisors of the free term, and lower the degree of the equation by dividing its left side by a linear binomial ( lambda-lambda_{1}) “by a corner” or according to Horner.