Again, here is the function:
$T:M_{2,2}\rightarrow R, T(A)=|A|$
I was able to prove that its not a linear transformation because $T(A+B) \neq T(A)+T(B)$ in fact, $T(A+B) = C$ where $C$ is a new matrix.
But I tried to prove $T(cu) = cT(u)$ and all I did was multiply a scalar by an arbitrary matrix and I got $c$ times each scalar and it seems to me that scalar multiplication is preserved. But the solution manual says:
T is not a linear transformation because it does not preserve addition nor scalar multiplication. For example, $T(I_2) = 1$ but $T(2I_2) = 4 \neq 2T(I_2)$.
I don't understand that solution at all. Proving the linear transformation aside, how is it that $T(I_2) = 1$? And, $T(2I_2) = 4$? Don't these just product matrices with elements change to the scalar multiple times the element and still produce a matrix? How can these show the results as being just a scalar?
Finally, I'd love to understand the solution and how scalar multiplication fails. Sorry if this seems obvious, I just don't understand.
Vertical bars mean determinant here. One has $T(2I_2)=\left|\begin{smallmatrix}2&0\\ 0&2\end{smallmatrix}\right|=4$.