Question

The rank of the matrix $A=\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 3& 4& 5\end{array}\right]$, is

• A. 1
• B. 2
• C. 3
• D. none of these

Answer 1

The correct option is B 2

$A=\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 3& 4& 5\end{array}\right]$
Here, $|A|=1-4+3=0$
So, rank of matrix $<3$.
Now, we will reduce into Echelon form,
$R_2\to R_2-4R_1$
$\left[\begin{array}{ccc}1& 2& 3\\ 0& -3& -6\\ 3& 4& 5\end{array}\right]$
$R_3\to R_3-3R_1$
$\left[\begin{array}{ccc}1& 2& 3\\ 0& -3& -6\\ 0& -2& -4\end{array}\right]$
$R_3\to R_3-\frac{2}{3}{R}_2$
$\left[\begin{array}{ccc}1& 2& 3\\ 0& -3& -6\\ 0& 0& 0\end{array}\right]$
$R_2\to -\frac{1}{3}{R}_2$
$\left[\begin{array}{ccc}1& 2& 3\\ 0& 1& 2\\ 0& 0& 0\end{array}\right]$
Here, there are 2 non-zero rows in the Echelon form of A.
Hence, rank =2