Determine the rank of the matrix $$\mathbf { A } = \left( \begin{array} { l l l l } 1 & - 1 & - 1 & 1
2 & - 1 & - 4 & 3
3 & - 3 & - 2 & 2
5 & - 4 & - 6 & 5 \end{array} \right)$$ Show that if $$\mathbf { A x } = p \left( \begin{array} { l } 1
2
3
5 \end{array} \right) + q \left( \begin{array} { l } - 1
- 1
- 3
- 4 \end{array} \right) + r \left( \begin{array} { l } - 1
- 4
- 2
- 6 \end{array} \right)$$ where \(p , q\) and \(r\) are given real numbers, then $$\mathbf { x } = \left( \begin{array} { c } p + \lambda
q + \lambda
r + \lambda
\lambda \end{array} \right) ,$$ where \(\lambda\) is real. Find the values of \(p , q\) and \(r\) such that $$p \left( \begin{array} { l } 1
2
3
5 \end{array} \right) + q \left( \begin{array} { l } - 1
- 1
- 3
- 4 \end{array} \right) + r \left( \begin{array} { l } - 1
- 4
- 2
- 6 \end{array} \right) = \left( \begin{array} { r } 3
7
8