How to efficiently use a calculator in a linear algebra exam, if allowed We are allowed to use a calculator in our linear algebra exam. Luckily, my calculator can also do matrix calculations.

Let's say there is a task like this:


  Calculate the rank of this matrix:
  
  $$M =\begin{pmatrix} 5 &amp; 6 &amp; 7\\  12 &amp;4  &amp;9 \\  1 &amp; 7 &amp; 4
\end{pmatrix}$$


The problem with this matrix is we cannot use the trick with multiples, we cannot see multiples on first glance and thus cannot say whether the vectors rows / columns are linearly in/dependent.
Using Gauss is also very time consuming (especially in case we don't get a zero line and keep trying harder).

Enough said, I took my calculator because we are allowed to use it and it gives me following results:

$$M =\begin{pmatrix} 1 &amp; 0{,}3333 &amp; 0{,}75\\  0 &amp;1  &amp;0{,}75 \\  0 &amp; 0 &amp; 1
\end{pmatrix}$$

I quickly see that $\text{rank(M)} = 3$ since there is no row full of zeroes.

Now my question is, how can I convince the teacher that I calculated it? If the task says "calculate" and I just write down the result, I don't think I will get all the points. What would you do?

And please give me some advice, this is really time consuming in an exam.
There is a very nice trick for showing that such matrix has full rank, it can be performed in a few seconds without any calculator or worrying "moral bending". The entries of $M$ are integers, so the determinant of $M$ is an integer, and $\det M\mod{2} = \det(M\mod{2})$. Since $M\pmod{2}$ has the following structure
$$ \begin{pmatrix} 1 &amp; 0 &amp; 1 \\ 0 &amp; 0 &amp; 1 \\ 1 &amp; 1 &amp; 0\end{pmatrix} $$
it is trivial that $\det M$ is an odd integer. In particular, $\det M\neq 0$ and $\text{rank}(M)=3$.
You're allowed to use your calculator.  So, if it were me on the test, I'd write something like this:


  $$
\pmatrix{5&amp;6&amp;7\\12&amp;4&amp;9\\1&amp;7&amp;4} \overset{REF}{\to} 
\pmatrix{
1 &amp; 0,3333 &amp; 0,75\\  0 &amp;1  &amp;0,75 \\  0 &amp; 0 &amp; 1
}
$$
  because the reduced form of $M$ has no zero rows, $M$ has rank $3$.


REF here stands for row-echelon form.



Note: You should check with your professor whether or not this constitutes a sufficient answer.  It may be the case that your professor wants any matrix-calculations to be done by hand.  See Robert Israel's comment.

If that's the case, then you should do row-reduction by hand.  It only takes 3 row operations, though.