Question

The value of p such that the vector \(\left[ {\begin{array}{*{20}{c}} 1\\ 2\\ 3 \end{array}} \right]{\rm{\;}}\)is an eigenvector of the matrix \(\left[ {\begin{array}{*{20}{c}} 4&1&2\\ {\rm{p}}&2&1\\ {14}&{ - 4}&{10} \end{array}} \right]\) is __________.

Answer 1

\(\begin{array}{l} {\rm{AX\;}} = {\rm{\;\lambda X}}\\ \left[ {\begin{array}{*{20}{c}} 4&1&2\\ {\rm{p}}&2&1\\ {14}&{ - 4}&{10} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} 1\\ 2\\ 3 \end{array}} \right] = {\rm{\lambda }}\left[ {\begin{array}{*{20}{c}} 1\\ 2\\ 3 \end{array}} \right]\\ \therefore {\rm{\;}}4{\rm{\;}} + {\rm{\;}}2{\rm{\;}} + {\rm{\;}}6{\rm{\;}} = {\rm{\;\lambda \;\;\;}} \Rightarrow {\rm{\;\lambda \;}} = {\rm{\;}}12\\ {\rm{p\;}} + {\rm{\;}}4{\rm{\;}} + {\rm{\;}}3{\rm{\;}} = {\rm{\;}}2{\rm{\lambda \;\;\;\;}} \Rightarrow {\rm{\;P\;}} = {\rm{\;}}24{\rm{\;}}-{\rm{\;}}7{\rm{\;}} = {\rm{\;}}17 \end{array}\)