7 The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) is represented by the matrix \(\mathbf { A }\), where
$$\mathbf { A } = \left( \begin{array} { r r r r }
1 & - 1 & - 2 & 3
5 & - 3 & - 4 & 25
6 & - 4 & - 6 & 28
7 & - 5 & - 8 & 31
\end{array} \right)$$
- Find the rank of \(\mathbf { A }\) and a basis for the null space of T .
- Find the matrix product \(\mathbf { A } \left( \begin{array} { r } - 1
1
- 1
1 \end{array} \right)\) and hence find the general solution of the equation \(\mathbf { A } \mathbf { x } = \left( \begin{array} { r } 3
21
24
27 \end{array} \right)\).
\(8 \quad\) Let \(I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \sec ^ { n } x \mathrm {~d} x\) for \(n > 0\).