6 The matrix \(\mathbf { M }\) is given by $$\mathbf { M } = \frac { 1 } { 10 } \left[ \begin{array} { c c c } a & a & - 6
0 & 10 & 0
9 & 14 & - 13 \end{array} \right]$$ where \(a\) is a real number. The vectors \(\mathbf { v } _ { 1 } , \mathbf { v } _ { 2 }\), and \(\mathbf { v } _ { 3 }\) are eigenvectors of \(\mathbf { M }\)
The corresponding eigenvalues are \(\lambda _ { 1 } , \lambda _ { 2 }\), and \(\lambda _ { 3 }\) respectively.
It is given that \(\lambda _ { 2 } = 1\) and \(\mathbf { v } _ { 1 } = \left[ \begin{array} { l } 1
0
3 \end{array} \right] , \mathbf { v } _ { 2 } = \left[ \begin{array} { l } 1
1
1 \end{array} \right]\) and \(\mathbf { v } _ { 3 } = \left[ \begin{array} { l } c
0
1 \end{array} \right]\),
where \(c\) is an integer. 6

1. 1. Find the value of \(\lambda _ { 1 }\)
      6
2. (ii) Find the value of \(a\)
   6
3. Find the integer \(c\) and the value of \(\lambda _ { 3 }\)
   6
4. Find matrices \(\mathbf { U } , \mathbf { D }\) and \(\mathbf { U } ^ { - 1 }\), such that \(\mathbf { D }\) is diagonal and \(\mathbf { M } = \mathbf { U D U } ^ { - 1 }\)