How can I use conventional math notation in Mathematica?

For example, instead of writing Det[Matrix imput], I would like to simply put the matrix inside 2 straight lines. Instead of writing Cos[x]^2, I would like to write Cos^2[x], with the square right after the Cos. And instead of writing the differential operator Dx[expression,x], I would like to write Dx[expression], without the x at the end, assuming implicitly the underscript on D means the variable the differentiation is occuring on.

I mean, I would like to be able to write mathematical equations and expressions in Mathematica in exactly the same way I write on a piece of paper. Could you help me?

You might try using a palette in the Palettes menu item.

Programmatically with many expressions you can try this:

TraditionalForm[HoldForm[Det[{{a, 8}, {b, 1}}]]]
TraditionalForm[Defer[Det[{{a, 8}, {b, 1}}]]]

ScreenShot2015-08-29at12.55.55AM.png

Read help on HoldForm and Defer and also take a look at Inactive which is useful in other situations. Also this blog could help:

Converting to Conventional Mathematical Typesetting

I see, now that I took a closer look at the pallete I found many symbols I didn’t see before

Is it possible to edit the palletes so as to include the TraditionalForm of the determinant?

Never mind. I found I can easily convert to Traditional Form using Shift+Ctrl+T. Nevertheless, how may the ambiguities arise if I’m using this form? Is there a source whichh might indicate where those ambiguities might occur and where they are troublesome?