Surely a $1\times 1$ matrix can only 'produce' vectors with 1 entry, and can take as input also only one entry vectors.
So, is there any use for $1\times 1$ matrices?
Since to me they do the same like scalars, only worse.
What is the use of the number zero? It doesn't do anything when you add it, and it always does the same thing when you multiply it.
Maybe the point of $1\times 1$ matrices is to not have to change the definition of matrix to exclude the $1\times 1$ case.
Maybe you come across a 1 dimension linear map in the wild and you want to make sure it's invertible and you've taken a linear algebra class and you don't want to draw or think about computing an inverse and you remember that the determinant of a 1 by 1 matrix is the scalar.
The point of a $n\times n$ (or $m\times n$)matrix is usually to generalize to $n$ dimensions what you already knew in $1$-dimension. And the point of the $1\times 1$ matrix is to give you back what you knew in $1$-dimension. So you won't need to prove something twice, once for 1-dimension and once for multiple dimensions (ಠ‿ಠ)