Specify fully the transformation T of the plane associated with the matrix \(\mathbf { M }\), where \(\mathbf { M } = \left( \begin{array} { l l } 1 & \lambda 0 & 1 \end{array} \right)\) and \(\lambda\) is a non-zero constant.
Find detM.
Deduce two properties of the transformation T from the value of detM.
Prove that \(\mathbf { M } ^ { n } = \left( \begin{array} { c c } 1 & n \lambda 0 & 1 \end{array} \right)\), where \(n\) is a positive integer.
Hence specify fully a single transformation which is equivalent to \(n\) applications of the transformation T.