if $p>n$, then the column vectors must be linearly dependent

If $p>n$, then the column vectors must be linearly independent

If $p=n$, then the column vectors must be orthogonal

If $p < n$, then the column vectors must be linearly independent

The value of the determinant of $A$ is either +1 or -1 .

The eigenvalues of $A$ have modulus 1.

$\|A x\|=\|x\|$, for all $x \in R^n$, where $\|x\|$ denotes the Euclidean norm of $x$, and $(A x)^{\top}(A y) \neq x^{\top} y$, for all distinct $x, y \in R^n$.

$\|A x\|=\|x\|$, for all $x \in R^n$, where $\|x\|$ denotes the Euclidean norm of $x$, and $(A x)^T(A y)=x^T y$, for all distinct $x, y \in R^n$.

$A^{-1}$ exists

The system of equations $A^m x=b$ also has a unique solution for $m=1,2,3, \ldots$

$\operatorname{rank}(\mathrm{A})=\operatorname{rank}\left(\mathrm{A}^{\mathrm{m}}\right)$, for $m=1,2,3, \ldots$

$\operatorname{rank}(A)<\operatorname{rank}([A \mid b])$, where $[A \mid b]$ denotes the augmented matrix.