  ### Home > PC3 > Chapter 9 > Lesson 9.2.2 > Problem9-116

9-116.

The linear transformation $M=\left[ \begin{array} { c c } { \cos ( 40 ^ \circ ) } & { \sin ( 40 ^ \circ ) } \\ { -\sin ( 40 ^ \circ ) } & { \cos ( 40 ^ \circ ) } \end{array} \right]$.

1. Algebraically describe the effect of $M$ on $(x, y)$.

$\left[ \begin{array} { c c } { \cos ( 40 ^ \circ ) } & { \sin ( 40 ^ \circ ) } \\ { -\sin ( 40 ^ \circ ) } & { \cos ( 40 ^ \circ ) } \end{array} \right] \left[ \begin{array} { c c} { x } \\{y} \end{array} \right]=$

2. Geometrically describe $M$.

Try transforming the points $(1, 0)$ and $(0, 1)$. Where do they end up in relation to where they started?

3. Write the matrix for $M^{−1}$.

$\left[ \begin{array} { c c } { \cos ( 40 ^ \circ ) } & { \sin ( 40 ^ \circ ) } \\ { -\sin ( 40 ^ \circ ) } & { \cos ( 40 ^ \circ ) } \end{array} \right] \left[ \begin{array} { c c} { a }&{b} \\{c} &{d}\end{array} \right]= \left[ \begin{array} { c c} { 0 }&{1} \\{1} & {0}\end{array} \right]$

Complete the multiplication and solve for $a, b, c,$ and $d$.