CAIE FP1 (Further Pure Mathematics 1) 2012 November

10 Write down the eigenvalues of the matrix \(\mathbf { A }\), where $$\mathbf { A } = \left( \begin{array} { r r r } 1 & 4 & - 16
0 & 2 & 3
0 & 0 & 3 \end{array} \right)$$ Find corresponding eigenvectors. Let \(n\) be a positive integer. Write down a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that $$\mathbf { A } ^ { n } = \mathbf { P D } \mathbf { P } ^ { - 1 }$$ Find \(\mathbf { P } ^ { - 1 }\) and \(\mathbf { A } ^ { n }\). Hence find \(\lim _ { n \rightarrow \infty } \left( 3 ^ { - n } \mathbf { A } ^ { n } \right)\).

The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 3 }\) is represented by the matrix \(\mathbf { M }\), where $$\mathbf { M } = \left( \begin{array} { r r r r } 2 & 1 & - 1 & 4
3 & 4 & 6 & 1
- 1 & 2 & 8 & - 7 \end{array} \right)$$ The range space of T is \(R\). In any order,

1. show that the dimension of \(R\) is 2 ,
2. find a basis for \(R\) and obtain a cartesian equation for \(R\),
3. find a basis for the null space of T . The vector \(\left( \begin{array} { l } 8
   7
   k \end{array} \right)\) belongs to \(R\). Find the value of \(k\) and, with this value of \(k\), find the general solution of $$\mathbf { M x } = \left( \begin{array} { l } 8
   7
   k \end{array} \right)$$