The rank of a matrix is the maximum number of its linearly independent column vectors (or row vectors).
It also can be shown that the columns (rows) of a square matrix are linearly independent only if the matrix is nonsingular. In other words, the rank of any nonsingular matrix of order n is n.
An $n × n$ matrix $A$ is called nonsingular or invertible if there exists an $n × n$ matrix $B$ such that
$AB=BA=I$.
If $A$ does not have an inverse, $A$ is called singular.
Definition. A matrix $A$ is orthogonal if $A^TA=AA^T=I$.
The range (also called the column space or image) of a $m × n$ matrix $A$ is the span (set of all possible linear combinations) of its column vectors. The column space of a matrix is the image or range of the corresponding matrix transformation.