Question

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

• A. One
• B. Two
• C. Three
• D. None of the above

Answer 1

The correct option is B Two

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

$R_2\to R_2+3R_1$

$R_3\to R_3-2R_1$

$\sim \left[\begin{array}{ccc}1& -2& -1\\ 0& -3& -3\\ 0& 6& 6\end{array}\right]$

$R_3\to R_3+2R_2$

$\sim \left[\begin{array}{ccc}1& -2& -1\\ 0& -3& -3\\ 0& 0& 0\end{array}\right]$

$\Rightarrow Rank\left(A\right)=2$

Alternative Solution:

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

= -1 (-6 -6) + 4 (3 - 6)

= 12 - 12 = 0

$M_{11}=\left|\begin{array}{cc}3& 0\\ 2& 4\end{array}\right|$ = 12 $\ne$ 0

$\Rightarrow Rank\left(A\right)=2$