Editing Rotation matrix (section)

== Multiplication ==
The inverse of a rotation matrix is its transpose, which is also a rotation matrix:
:<math>\begin{align} \left(Q^\mathsf{T}\right)^\mathsf{T} \left(Q^\mathsf{T}\right) &= Q Q^\mathsf{T} = I\\ \det Q^\mathsf{T} &= \det Q = +1. \end{align}</math>
The product of two rotation matrices is a rotation matrix:
:<math>\begin{align}
 \left(Q_1 Q_2\right)^\mathsf{T} \left(Q_1 Q_2\right) &= Q_2^\mathsf{T} \left(Q_1^\mathsf{T} Q_1\right) Q_2 = I \\
 \det \left(Q_1 Q_2\right) &= \left(\det Q_1\right) \left(\det Q_2\right) = +1.
\end{align}</math>
For {{math|''n'' > 2}}, multiplication of {{math|''n'' × ''n''}} rotation matrices is generally not [[commutative]].
:<math>\begin{align}
Q_1 &= \begin{bmatrix}0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1\end{bmatrix} &
Q_2 &= \begin{bmatrix}0 & 0 & 1 \\ 0 & 1 & 0 \\ -1 & 0 & 0\end{bmatrix} \\
Q_1 Q_2 &= \begin{bmatrix}0 & -1 & 0 \\ 0 & 0 & 1 \\ -1 & 0 & 0\end{bmatrix} &
Q_2 Q_1 &= \begin{bmatrix}0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0\end{bmatrix}.
\end{align}</math>
Noting that any [[identity matrix]] is a rotation matrix, and that matrix multiplication is [[associative]], we may summarize all these properties by saying that the {{math|''n'' × ''n''}} rotation matrices form a [[group (mathematics)|group]], which for {{math|''n'' > 2}} is [[nonabelian group|non-abelian]], called a [[special orthogonal group]], and denoted by {{math|SO(''n'')}}, {{math|SO(''n'','''R''')}}, {{math|SO<sub>''n''</sub>}}, or {{math|SO<sub>''n''</sub>('''R''')}}, the group of {{math|''n'' × ''n''}} rotation matrices is isomorphic to the group of rotations in an {{nowrap|{{math|''n''}}-dimensional}} space. This means that multiplication of rotation matrices corresponds to composition of rotations, applied in left-to-right order of their corresponding matrices.