what is a rank one matrix?

Answer 1

It means that the columns of the matrix are scalar multiples of each other.

Answer 2

The column rank of a matrix A is the maximal number of linearly independent columns of A. Likewise, the row rank is the maximal number of linearly independent rows of A.

The column rank and the row rank are always equal thus they are simply called the rank of A. It is commonly denoted by either rk(A) or rank A.

The maximal number of linearly independent columns of the m-by-n matrix A with entries in the field F is equal to the dimension of the column space of A (the column space being the subspace of Fm generated by the columns of A). Since the column rank and the row rank are the same, we can also define the rank of A as the dimension of the row space of A.

If one considers the matrix A as a linear map.

Answer 3

It can be defined as the number of linearly independent rows or columns of the matrix considred as vectors in a vector(linear space)