If the points `(x_1,y_1),(x_2,y_2)and(x_3,y_3)` are collinear, then the rank of the matrix `{:[(x_1,y_1,1),(x_2,y_2,1),(x_3,y_3,1)]:}` will always be less than

Options

A. 3
B. 2
C. 1
D. none of these

Answer

To solve the problem, we need to understand the relationship between collinear points and the rank of the matrix formed by these points.

### Step-by-Step Solution:

1. **Understanding Collinearity**:
- Three points \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) are said to be collinear if they lie on the same straight line.

2. **Forming the Matrix**:
- We can represent these points in a matrix form as follows:
\[ A = \begin{bmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{bmatrix} \]

3. **Calculating the Determinant**:
- The determinant of this matrix \(A\) can be calculated using the formula for the determinant of a 3x3 matrix:
\[ \text{det}(A) = x_1(y_2 \cdot 1 - y_3 \cdot 1) - y_1(x_2 \cdot 1 - x_3 \cdot 1) + 1(x_2y_3 - x_3y_2) \]
- However, for collinear points, the determinant will always equal zero:
\[ \text{det}(A) = 0 \]

4. **Implications of the Determinant**:
- Since the determinant is zero, this indicates that the rows of the matrix \(A\) are linearly dependent.

5. **Rank of the Matrix**:
- The rank of a matrix is defined as the maximum number of linearly independent row vectors in the matrix.
- For a matrix with three rows, if the determinant is zero, the rank must be less than the number of rows. Therefore, the rank of matrix \(A\) must be less than 3.

6. **Conclusion**:
- Hence, we conclude that if the points \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) are collinear, the rank of the matrix \(A\) will always be less than 3.

### Final Answer:
The rank of the matrix will always be less than 3.

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