- \(\mathbf { M } = \left( \begin{array} { r r r } 0 & 1 & 9
1 & 4 & k
1 & 0 & - 3 \end{array} \right)\), where \(k\) is a constant.
Given that \(\left( \begin{array} { r } 7
19
1 \end{array} \right)\) is an eigenvector of the matrix \(\mathbf { M }\),
- find the eigenvalue of \(\mathbf { M }\) corresponding to \(\left( \begin{array} { r } 7
19
1 \end{array} \right)\), - show that \(k = - 7\)
- find the other two eigenvalues of the matrix \(\mathbf { M }\).
The image of the vector \(\left( \begin{array} { c } p
q
r \end{array} \right)\) under the transformation represented by \(\mathbf { M }\) is \(\left( \begin{array} { r } - 6
21
5 \end{array} \right)\). - Find the values of the constants \(p , q\) and \(r\).