Problem

7. $A = \begin{pmatrix} -\dfrac{3}{2} & -\dfrac{1}{2} \\ \dfrac{1}{2} & -\dfrac{3}{2} \end{pmatrix}$

(a) Determine the matrix $A^2$

(b) Describe fully the single geometrical transformation represented by the matrix $A^2$

(c) Hence determine the smallest positive integer value of $n$ for which $A^n = I$

The matrix $B$ represents a stretch scale factor $4$ parallel to the $x$-axis.

(d) Write down the matrix $B$

The transformation represented by matrix $A$ followed by the transformation represented by matrix $B$ is represented by the matrix $C$

(e) Determine the matrix $C$

The parallelogram $P$ is transformed onto the parallelogram $P'$ by the matrix $C$

(f) Given that the area of parallelogram $P'$ is $20$ square units, determine the area of parallelogram $P$