5 By using the determinant of an appropriate matrix, find the values of \(\lambda\) for which the simultaneous equations
$$\begin{array} { r }
3 x + 2 y + 4 z = 5
\lambda y + z = 1
x + \lambda y + \lambda z = 4
\end{array}$$
do not have a unique solution for \(x , y\) and \(z\).
\includegraphics[max width=\textwidth, alt={}, center]{f074de40-08b6-47a6-a0d2-d3cbe628cacc-3_556_759_233_653}
The diagram shows the unit square \(O A B C\), and its image \(O A B ^ { \prime } C ^ { \prime }\) after a transformation. The points have the following coordinates: \(A ( 1,0 ) , B ( 1,1 ) , C ( 0,1 ) , B ^ { \prime } ( 3,2 )\) and \(C ^ { \prime } ( 2,2 )\).
- Write down the matrix, \(\mathbf { X }\), for this transformation.
- The transformation represented by \(\mathbf { X }\) is equivalent to a transformation P followed by a transformation Q. Give geometrical descriptions of a pair of possible transformations P and Q and state the matrices that represent them.
- Find the matrix that represents transformation Q followed by transformation P .