Question

If the points $(x_1,y_1);(x_2,y_2);(x_3,y_3)$ are collinear and $A=$ $\left\{\begin{array}{ccc}{x}_1& y_1& 1\\ x_2& y_2& 1\\ x_3& y_3& 1\end{array}\right\}$ then the rank (A) is

• A. $=3$
• B. $<3$
• C. $>3$
• D. $>4$

Answer 1

Answer: (B)

$<3$

We know that area of triangle with vertices $(x_1,y_1);(x_2,y_2);(x_3,y_3)$ is given by
$A=$ $\frac{1}{2}\left|\left\{\begin{array}{ccc}{x}_1& y_1& 1\\ x_2& y_2& 1\\ x_3& y_3& 1\end{array}\right\}\right|$
Now since points are collinear $\Rightarrow$ Area of triangle will vanish
$\Rightarrow \left\{\begin{array}{ccc}{x}_1& y_1& 1\\ x_2& y_2& 1\\ x_3& y_3& 1\end{array}\right\}=0$
$\Rightarrow A$ is rank defficient i.e. rank$\left(A\right)<3$