Concept:
If matrix is upper triangular matrix or lower triangular matrix then eigen value of that matrix is diagonal element.
\(A = \left[ {\begin{array}{*{20}{c}} a&d&e\\ o&b&f\\ o&o&c \end{array}} \right]\;B = \left[ {\begin{array}{*{20}{c}} a&o&o\\ d&b&o\\ e&f&c \end{array}} \right]\)
A and B are upper triangular matrix and lower triangular matrix respectively and its eigen value is a, b, c.
Calculation:
\(A = \left[ {\begin{array}{*{20}{c}} 2&1&0\\ 0&2&0\\ 0&0&3 \end{array}} \right]\)
Above matrix is upper triangular matrix so the eigen value of matrix A is 2, 2, 3.
As the two Eigen value of given matrix are identical, the eigenvectors resulting from these identical eigen values would be identical or linearly dependent.
Therefore, number of linearly independent eigen vectors (since two eigen values are identical)
= n - 1
= 3 - 1
= 2
