I'm revising some elementary linear algebra after a multi-decade break in which I've forgotten most of it. I've taken a look at a few introductory text books, and there seems to be a common line of argument about singular matrices that seems to me to be wrong: many of these textbooks claim that because the determinant of a singular matrix (let's say a $2 \times 2$ matrix over the reals) can be viewed as "area destroying" then the matrix maps $\mathbb{R}^2$ to $\mathbb{R}$, and it's therefore "obvious" that it cannot be inverted.
Surely this is twaddle since the cardinality of $\mathbb{R}^2$ is the same as that of $\mathbb{R}$ ? Isn't the real argument to show that a singular 2x2 matrix is not injective ?
Who's confused here? Me or the books?