Linearity / Matrix functions?

I shouldn't have left it so late to ask this, but the answers to any of these would be most helpful...

Is the function mapping a matrix A to the determinant of A linear?

I'm thinking it's non-linear, but simply because I can't think of a transformation matrix that could map [[a,b],[c,d]] <--- (A 2x2 matrix a, b, c, d reading left to right... apologies for absence of Latex!) to ad-bc...

**Is the operation mapping f to f''+3f' linear? **

**(Where f is over the collection of all infinitely differentiable functions)**

I know differentiation is linear, but does it hold for two separate derivatives added together?

Fine, A(cx + dy) = c(Ax)+d(Ay) if and only if the function is linear, but how does one actually go about picking these x and y s, or creating a matrix A to test one's theory?

**Is a function mapping f to its second derivative linear?**

Aye, I'm presuming this is linear, since differentiation is linear (although I'm not sure how to explain that...)

**Is the function mapping a matrix A to its trace linear?**

This one is linear, right? Because you're just adding two entries together. Again, I'm not sure how to construct the function as a matrix...

**Is the function mapping x to 3x + 2 linear?**

This is a straight out 'no', right? 3x is fine, but you can't just add a constant like that, right?

**Let C = [[1,2],[3,4]]. Is the function mapping A to AC-CA linear?**

Again, I'm guessing it's not, for similar reasons to the previous...

Any help on any of these much appreciated.

Thanks in advance...