Here is one type of non-standard question which you might try to sell in a sort of puzzle style.
Can you use the quadratic formula to solve the cubic equation $x^3-26x-5=0$?
The answer obviously should be yes, the trick is to write the equation as $$(-x)5^2-5+(x^3-x)=0.$$
Viewing this a quadratic equation in '$5$', and using the quadratic formula, we get that $5=\frac{1\pm \sqrt{1+4x(x^3-x)}}{-2x}$.
Thus $$-10x=1\pm \sqrt{4x^4-4x^2+1}=1\pm \sqrt{(2x^2-1)^2}.$$
Thus $$-10x-1=\pm(2x^2-1).$$
Now this is a quadratic equation and thus can be solved. Now you have two zeroes, and the third can be found by factoring out the other two. In fact you can do this trick for equations of the form $x^3-ax-b=0$ as long as the expression underneath the root is nice.
But then again, there is nothing real-life about this question. It is somewhat surprising to see that the quadratic formula can sometimes be used to find roots of cubics. So in that respect it might be interesting.
I have published nearly $2500$ pages of books, scholarly papers, manuals, etc., in $\mathrm{\LaTeX}$ (and many many more in memos, internal notes, class notes, etc.). (This doesn't count the current $420$ pages of my next book, entirely in $\mathrm{\LaTeX}$.) $\mathrm{\LaTeX}$ is by far better than any other typesetting software for technical publishing. That is why it is the preferred, or one of the preferred, default standards for the American Mathematical Society, IEEE, and as far as I know every technical society. One trick, though, is to use Mathematica and its user-friendly templates, for instance for matrices, vectors, and such. Then convert it to $\mathrm{\LaTeX}$ source by TeXForm[...]. But I recommend just getting fluent in $\mathrm{\LaTeX}$. It will also help with MathJax on this site.
Three "bonus benefits" are
- It is portable: you can email your $\mathrm{\LaTeX}$ source to a co-author who uses a different operating system, different computer, even different paper size, and you can easily collaborate.
- It is free. There are lots of free versions available.
- It is beautiful... the type, the layout, is so lovely and clear.
I like the following analogy. If you just need to drive across town to shop, a simple car (automatic, inexpensive, easy to learn) will suffice. But if you're a power driver and need to go fast and take lots of sharp corners and hills, get a Ferrari. True, it takes more time to learn, but if you want to do a lot, it is better.
Same thing with other simple typesetting compared to $\mathrm{\LaTeX}$.
Best Answer
If you are going to teach your students linear algebra, then you should limit yourself to teaching linear algebra.
If you absolutely have to use computers (which is itself an unnecessary and harmful distraction in the process of teaching mathematics), I believe that you should use software that is easy to use and allows your students to concentrate on learning mathematics.
If you will decide to use some fancy programming language, then you should keep in mind that your students will be struggling with both learning linear algebra, and learning new programming environment.
So imho:
1)Best solution: do not use computers!!! People will really learn something only when they will do all calculations by hand. This is the only way to understand all basic concepts of linear algebra. There is no king's road to mathematics.
2)Poor solution: use some user-friendly and easy-to-learn CAS (or numeric program), like Matlab or Mathematica or Octave. At least people will learn something about linear algebra.
3)Educational disaster: use some fancy programming language, like Python. People will learn almost nothing about linear algebra and almost nothing about programming. Moreover: they will associate linear algebra with programming, mixing concepts from both fields.