Find the matrix that represents a shear with the \(y\)-axis invariant, the image of the point \(( 1,0 )\) being the point \(( 1,4 )\).
The matrix \(\mathbf { X }\) is given by \(\mathbf { X } = \left( \begin{array} { r r } \frac { 1 } { 2 } \sqrt { 2 } & \frac { 1 } { 2 } \sqrt { 2 } - \frac { 1 } { 2 } \sqrt { 2 } & \frac { 1 } { 2 } \sqrt { 2 } \end{array} \right)\).
Describe fully the geometrical transformation represented by \(\mathbf { X }\).
Find the value of the determinant of \(\mathbf { X }\) and describe briefly how this value relates to the transformation represented by \(\mathbf { X }\).