Question

The rank of the matrix $A=\left[\begin{array}{ccc}1& 2& 3\\ 2& 3& 4\\ 0& 1& 2\end{array}\right]$ is:

• A. $3$
• B. $2$
• C. $1$
• D. $0$

Answer 1

The correct option is B $2$

Given,

$\left(\begin{array}{ccc}1& 2& 3\\ 2& 3& 4\\ 0& 1& 2\end{array}\right)$

$R_1\leftrightarrow R_2$

$R_2\to R_2-\frac{1}{2}{R}_1$

$=\left(\begin{array}{ccc}2& 3& 4\\ 0& \frac{1}{2}& 1\\ 0& 1& 2\end{array}\right)$

$R_2\leftrightarrow R_3$

$R_3\to R_3-\frac{1}{2}{R}_2$

$=\left(\begin{array}{ccc}2& 3& 4\\ 0& 1& 2\\ 0& 0& 0\end{array}\right)$

$R_1\to R_1-3R_2$

$=\left(\begin{array}{ccc}2& 0& -2\\ 0& 1& 2\\ 0& 0& 0\end{array}\right)$

$R_1\to \frac{1}{2}{R}_1$

$=\left(\begin{array}{ccc}1& 0& -1\\ 0& 1& 2\\ 0& 0& 0\end{array}\right)$

rank $=2$