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
$\begin{bmatrix}x_{_1}&y_{_1}&1\\ x_{_2}&y_{_2}&1\\ x_{_3}&y_{_3}&1\end{bmatrix}$will always be less than

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

Answer 1

Answer: (B)

The given matrix is $\begin{bmatrix}x_{_1}&y_{_1}&1\\ x_{_2}&y_{_2}&1\\ x_{_3}&y_{_3}&1\end{bmatrix}$
using $R_{_2}\to R_{_2}-R_{_1}, R_{_3} \to R_{_3}-R_{_1}$
$\Delta=\begin{bmatrix} x_{_1 } & y_{_1} &1\\ x_{_2}-x_{_1}&y_{_2}-y_{_1}&0\\ x_{_3}-x_{_1}&y_{_3}-y_{_1}&0\end{bmatrix}=0$
$(\because$ points are collinear i.e., area of triangle =0 )
$\Rightarrow \begin{vmatrix}x_{_2}-x_{_1}&y_{_2}-y_{_1}\\ x_{_3}-x_{_1}&y_{_3}-y_{_1}\end{vmatrix}=0$
So, the rank of matrix is always less than $2$.