### Home > PC > Chapter 12 > Lesson 12.2.2 > Problem12-52

12-52.

The linear transformation $M$ is described by $\left[ \begin{array} { c c } { \operatorname { cos } 40 ^ { \circ } } & { \operatorname { sin } 40 ^ { \circ } } \\ { - \operatorname { sin } 40 ^ { \circ } } & { \operatorname { cos } 40 ^ { \circ } } \end{array} \right]$.

1. Describe the effect of $M$ on a point $p = \left(x, y\right)$.

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

2. Geometrically describe $M$.

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

3. Find $M ^{−1}$.

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

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