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

The correct option is A 3

Given, the points $(x_1,y_1),(x_2,y_2)$ and $(x_3,y_3)$ are collinear .
$\Rightarrow \left|\begin{array}{ccc}{x}_1& y_1& 1\\ x_2& y_2& 1\\ x_3& y_3& 1\end{array}\right|=0$ ....(1)

Now given, $A=\left[\begin{array}{ccc}{x}_1& y_1& 1\\ x_2& y_2& 1\\ x_3& y_3& 1\end{array}\right]$
Since,$|A|=0\left(by1\right)$
Thus, rank of A$<$order of matrix.
Hence, $r<3$