# Updating the inverse of a matrix

To understand how the axis and the angle control a rotation, let's do a small experiment.Put your thumb up against your monitor and try rotating your hand around it.Can you guess from the multiplication overview what the matrix should look like to translate a vector by [\begin \color1 & \color0 & \color0 & \color X \ \color0 & \color1 & \color0 & \color Y \ \color0 & \color0 & \color1 & \color Z \ \color0 & \color0 & \color0 & \color1 \end \cdot \begin x \ y \ z \ 1 \end = \begin x \color X\cdot 1 \ y \color Y\cdot 1 \ z \color Z\cdot 1 \ 1 \end] value a translation wouldn't have been possible.A scale transformation scales each of a vector's components by a (different) scalar.

For those among you who aren't very math savvy, the dot is a multiplication sign.Objects can be rotated around any given axis, but for now only the X, Y and Z axis are important.You'll see later in this chapter that any rotation axis can be established by rotating around the X, Y and Z axis simultaneously.It is commonly used to shrink or stretch a vector as demonstrated below.If you understand how the previous matrix was formed, it should not be difficult to come up with a matrix that scales a given vector by [\begin \color & \color0 & \color0 & \color0 \ \color0 & \color & \color0 & \color0 \ \color0 & \color0 & \color & \color0 \ \color0 & \color0 & \color0 & \color1 \end \cdot \begin x \ y \ z \ 1 \end = \begin \color\cdot x \ \color\cdot y \ \color\cdot z \ 1 \end].