Can a matrix be just 1 number? Can a matrix be just one number? Eg. Does 2 count as a matrix, if the question asks for a matrix to fit the question, but only the number 2 multiplying this particular matrix gives me the desired result?
Thanks
 A: Yes, a matrix with only one entry $c$, i.e., a $1 \times 1$ matrix, is equivalent to the scalar $c$. So any number in the underlying field can be thought of as a matrix with just one row, one column, i.e. one entry.
$$\begin{bmatrix} c\end{bmatrix} = c$$
A: It seems that your problem is really a different one. It seems that you want to ask:

Can the map $\mathbb R^n\to\mathbb R^n$, $x\mapsto \lambda x$, where $\lambda\in\mathbb R$, also be expressed by a matrix $A$, i.e. is there a matrix $A$ such that $Ax=\lambda x$ for all $x\in\mathbb R^n$?

The answer to this is:

Yes, just let $A=\lambda I$, where $I$ is the unit matrix (all diagonal entries $1$, all other entries $0$).

If you already know what a linear map is, then one could also say:

Yes, since $x\mapsto\lambda x$ is linear and each linear map can be represented by a matrix.

And as a hint for your other problem: There are to linear maps that map the square spanned by the vectors $(0,1)$ and $(1,0)$ to itself, namely the identity and the reflection by the main diagonal.
A: Yes, a number can be thought of as $1\times 1$ matrix. Note that for a general matrix $M$ (over, say, the real numbers) and a real number $a$, the notation $a.M$ is usually defined as the product $Ia. M$ where $I$ is the identity matrix, i.e. $a.M$ is obtained by multiplying all entries of $M$ with $a$.
