Find the rank of the matrix 1 4 - 1 2 3 0 0 1 2 .

Consider the matrix, $\left[\begin{array}{ccc}1& 4& -1\\ 2& 3& 0\\ 0& 1& 2\end{array}\right]$

$A$ is the matrix of the order $3\times 3$

Now,

$\left|A\right|=\left|\begin{array}{ccc}1& 4& -1\\ 2& 3& 0\\ 0& 1& 2\end{array}\right|$

$=1\left(6-0\right)-4\left(4-0\right)-1\left(2-0\right)$

$$\begin{gathered} =6-16-2 \\ =-12\ne 0 \end{gathered}$$

$\Rightarrow \left|A\right|\ne 0$

Rank of matrix $A=$Order of matrix$=3$

Rank of given matrix$=3$.