OK, here is what I get when I run the code posted above:

(let's see if this looks better)

Attachments:

Trig functions (most proggies solve using a reiman sum or built-in math in intel/amd/arm chip) may yield numeric results.

But many Mathematica facilities are aware of Trignomic Identities, and can keep arbitrary precision because they dont' use "intel math", but the symbolic math solving rules.

yes you know calculations using 0.45 will be inaccurate if done continually (see the opener in any free physics book or lab book about calculations involving precision and accuracy)

4/5 + 5/6 is done by the Kernel without ever digitizing numbers, it does this whenever possible, it's called "arbitrary precision numberic support" and most mathematic products (ie, MathCad) cannot even do it (Mathematica has always done it)

If you can, always use rational numbers (fractions) in calculations, use them until it's time to Print[] the number (view it, or let Graph chop/digitize it as it needs to). the old mathematica book suggested that, i'm hoping the new one still does.

it's not a worry until you see "0.04" in a book and your career might depend on knowing if that 4 is in an estimated last digit, if rounded from one, or is fully known (guess digit omitted), and if anyone knows what the next digits is :)

on the other hand, some Mathematica functions (you need to know which) may not give you an arbitrary precision result (ie, the approximation functions), but when they do it they keep the precision you ask for (which may be quite large, but if doing astronomical eqn, may need to be quite large to cover losses during calculations)

Hi Irenzo,

could you please explain what do you mean with "mat has still the approximation"?

The approximate number 0.45 is introduced in the input, and therefore produces an approximate number in the output. Compare to the code below, which uses the exact number 45/100. However, an approximate number is a number, and does nothing to interfere with multiplication, matrix or otherwise.

In[1]:= f1 = s Cos[\[Theta]1] + l2 Cos[\[Theta]2] + l3 Cos[\[Theta]3];

In[2]:= f2 = s Sin[\[Theta]1] + l2 Sin[\[Theta]2] + l3 Sin[\[Theta]3];

In[3]:= (jmat = {{D[f1, s], D[f1, \[Theta]3]}, {D[f2, s], 
      D[f2, \[Theta]3]}}) // MatrixForm ;

In[4]:= ddet = Det[jmat] ;

In[5]:= Simplify[ddet /. {l3 -> 45/100, \[Theta]1 -> \[Pi]}] ;

In[6]:= mat = 
 Simplify[Inverse[jmat] /. {l3 -> 45/100, \[Theta]1 -> \[Pi]}]


Out[6]= {{-1, -Tan[\[Theta]3]}, {0, (20 Sec[\[Theta]3])/9}}

Thank you Michael for your post. Why the matrix "mat" still has the apporoximation?