TITLE: How to calculate the rank of a matrix?
QUESTION [10 upvotes]: I need to calculate the rank of the matrix $A$, shown below:
  $$ 
A=
\begin{bmatrix}
3 & 2 & -1\\
2 & -3 & -5\\
-1 & -4 &- 3
\end{bmatrix}
$$


I know that I need to calculate $\det(A)$ and if $\det(A) \neq 0$ then the rank will be equal to $3$, but in this case I'm required to zero-out first column of matrix $A$ using element $a_{31} = -1$.

REPLY [12 votes]: Simply use row reduction: the rank is the number of non-zero rows after you've performed row reduction:
\begin{align}
&\begin{bmatrix}
3&2&-1\\2&-3&-5\\-1&-4&-3
\end{bmatrix}\rightsquigarrow
\begin{bmatrix}
1&4&3\\3&2&-1\\2&-3&-5
\end{bmatrix}\rightsquigarrow
\begin{bmatrix}
1&4&3\\0&-10&-10\\0&-11&-11
\end{bmatrix}\\[1ex]
\rightsquigarrow&\begin{bmatrix}
1&4&3\\0&1&1\\0&-11&-11
\end{bmatrix}\rightsquigarrow
\begin{bmatrix}
1&4&3\\0&1&1\\0&0&0
\end{bmatrix}
\end{align}
Thus, the rank is $2$.
Note that it was obvious after the second step.