Characterize a rotation matrix

Question

Given a matrix $A\in M_{2 \times 2}(\mathbb R)$ or $M_{3\times 3}(\mathbb R)$ how to determine if it is a rotation matrix? Is there any theorem that characterize a rotation matrix just by looking at columns, rows, or determinant of the given matrix $A$?

Answer 1

in two dimension, a rotation matrix cannot have a real eigenvalue on top of preserving the lengths. that is an orthogonal matrix $AA^\top = I.$

in three dimensions, $A$ must have an eigenvalue $1$ the dimension of $null(A-I) = 1$ and you always need $AA^\top = I$ to preserve the length.

Answer 2

In dimensions $2$ and $3$, a rotation matrix (around the origin, or an axis through the origin in dimension $3$) is an orthogonal matrix with determinant $1$.

An orthogonal matrix $A$ of dimension $n$ is a matrix in $M_n(\mathbf R)$ such that its inverse is its transpose: $$A\,^{\mathrm t\!}A=I,\enspace\text{so that}\enspace(\det A)^2=1$$

In dimension $3$, the rotation axis is the subspace of fixed points of $A$, i. e. the solutions of: $$(A-I)\cdot x=0.$$