Question

Find the rank of the matrix $\left|\begin{array}{cccc}0& 1& 2& 1\\ 2& -3& 0& -1\\ 1& 1& -1& 0\end{array}\right|$

Answer 1

Let $A=\left|\begin{array}{cccc}0& 1& 2& 1\\ 2& -3& 0& -1\\ 1& 1& -1& 0\end{array}\right|$
$\sim \left[\begin{array}{cccc}0& 1& 2& 1\\ 2& -2& 2& 0\\ 1& 1& -1& 0\end{array}\right]R_2⟶R_2+R_1$
$\sim \left[\begin{array}{cccc}0& 1& 2& 1\\ 1& -1& 1& 0\\ 1& 1& -1& 0\end{array}\right]R_2⟶\frac{1}{2}{R}_2$
$\sim \left[\begin{array}{cccc}0& 1& 2& 1\\ 1& -1& 1& 0\\ 2& 0& 0& 0\end{array}\right]R_3\to R_3+R_2$
The last equivalent matrix is in echelon form
The number of non-zero rows in this matrix is three
$\therefore$ $\rho \left(A\right)=3$