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:
$$M =\begin{pmatrix} 5 & 6 & 7\\ 12 &4 &9 \\ 1 & 7 & 4
\end{pmatrix}$$
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:
$$M =\begin{pmatrix} 1 & 0{,}3333 & 0{,}75\\ 0 &1 &0{,}75 \\ 0 & 0 & 1
\end{pmatrix}$$
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.
Best Answer
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 $$ \begin{pmatrix} 1 & 0 & 1 \\ 0 & 0 & 1 \\ 1 & 1 & 0\end{pmatrix} $$ it is trivial that $\det M$ is an odd integer. In particular, $\det M\neq 0$ and $\text{rank}(M)=3$.