Understand and use the relationship between the determinant and the effect of a linear transformation on area.

A linear transformation \( T \) is represented by the matrix \( M = \begin{pmatrix} 3 & 1 \\ -2 & 4 \end{pmatrix} \). A parallelogram in the \(xy\)-plane has area 15 square units. What is the area (in square units) of the image of this parallelogram under the transformation \( T \)?

A linear transformation \(T\) is defined by the matrix \(M = \begin{pmatrix} 3 & 1 \\ 2k & 4 \end{pmatrix}\), where \(k > 6\). The table below shows the vertices of a triangular region \(R\) and the corresponding image vertices after transformation \(T\). Given that the area of the image region is \(45\) square units, determine the value of \(k\).

A linear transformation is defined by the matrix \( M = \begin{pmatrix} 3 & 1 \\ 2 & 4 \end{pmatrix} \). (a) Calculate the determinant of \( M \). (1 mark) (b) A region in the plane has an area of 8 square units. Determine the area of the region after it is transformed by the matrix \( M \). (1 mark) (c) State the relationship that justifies your calculation in part (b). (1 mark)