Question: Use Exerise 27 to complete the proof of Theorem 1 for the case when A is lower triangular.

Let the matrix A is given as:

\(A = \left( {\begin{array}{*{20}{c}}{{a_{11}}}&{{a_{12}}}&{{a_{13}}}\\0&{{a_{22}}}&{{a_{23}}}\\0&0&{{a_{33}}}\end{array}} \right)\)

As \(\lambda \) is the eigenvalue of matrix A, then;

\(\begin{array}{c}\left| {A - \lambda I} \right| = 0\\\left( {\begin{array}{*{20}{c}}{{a_{11}} - \lambda }&{{a_{12}}}&{{a_{13}}}\\0&{{a_{22}} - \lambda }&{{a_{23}}}\\0&{{a_{23}}}&{{a_{33}} - \lambda }\end{array}} \right) = 0\\\left( {{a_{11}} - \lambda } \right)\left( {\left( {{a_{22}} - \lambda } \right)\left( {{a_{33}} - \lambda } \right) - 0} \right) - {a_{12}}\left( {0 - 0} \right) + {a_{13}}\left( {0 - 0} \right) = 0\\\lambda = {a_{11}},\,{a_{22}},\,{a_{33}}\end{array}\)

So, the diagonal elements are the eigenvalues of matrix A.

Hence, Theorem 1 has been proved.