Question

If the points $(x_1,y_1),(x_2,y_2)$ and $(x_3,y_3)$ are collinear, then the rank of the matrix $\left[\begin{array}{ccc}{x}_1& y_1& 1\\ x_2& y_2& 1\\ x_3& y_3& 1\end{array}\right]$ will always be less than

• A. $3$
• B. $2$
• C. $1$
• D. None of these

Answer 1

Answer: (B)

$2$

The given matrix is $\left[\begin{array}{ccc}{x}_1& y_1& 1\\ x_2& y_2& 1\\ x_3& y_3& 1\end{array}\right]$, using $R_2\to R_2-R_1,R_3\to R_3-R_1$
$△=\left[\begin{array}{ccc}{x}_1& y_1& 1\\ x_2-x_1& y_2-y_1& 0\\ x_3-x_1& y_3-x_1& 0\end{array}\right]$
($\because$ points are collinear i.e., area of triangle $=0$)
$\Rightarrow \left|\begin{array}{cc}{x}_2-x_1& y_2-y_1\\ x_3-x_1& y_3-y_1\end{array}\right|=0$
So, the rank of matrix is always less than $2$.