`A=[{:(l_(1),m_(1),n_(1)),(l_(2),m_(2),n_(2)),(l_(3),m_(3),n_(3)):}]` and `B=[{:(p_(1),q_(1),r_(1)),(p_(2),q_(2),r_(2)),(p_(3),q_(3),r_(3)):}]`
Where `p_(i), q_(i),r_(i)` are the co-factors of the elements `l_(i), m_(i), n_(i)` for `i=1,2,3`. If `(l_(1),m_(1),n_(1)),(l_(2),m_(2),n_(2))` and `(l_(3),m_(3),n_(3))` are the direction cosines of three mutually perpendicular lines then `(p_(1),q_(1), r_(1)),(p_(2),q_(2),r_(2))` and `(p_(3),q_(),r_(3))` are

Options

A. the direction cosines of three mutually perpendicular lines
B. the direction ratios of three mutually perpendicular lines which are not direction cosines.
C. the direction cosines of three lines which need not be perpendicular
D. the direction of three lines which need not be perpendicular

Explanation

To solve the problem, we need to establish the relationship between matrices A and B, given that the direction cosines in A represent three mutually perpendicular lines.

### Step-by-step Solution:

1. **Define the Vectors**:
Let the vectors corresponding to the direction cosines be defined as:
\[ \vec{A} = l_1 \hat{i} + m_1 \hat{j} + n_1 \hat{k} \]
\[ \vec{B} = l_2 \hat{i} + m_2 \hat{j} + n_2 \hat{k} \]
\[ \vec{C} = l_3 \hat{i} + m_3 \hat{j} + n_3 \hat{k} \]

2. **Understanding Co-factors**:
The co-factors of the elements in A are given as:
\[ p_1 = m_2 n_3 - m_3 n_2, \quad q_1 = n_2 l_3 - n_3 l_2, \quad r_1 = l_2 m_3 - l_3 m_2 \]
Similarly for \( p_2, q_2, r_2 \) and \( p_3, q_3, r_3 \).

3. **Condition of Perpendicularity**:
Since the direction cosines are mutually perpendicular, the scalar triple product of the vectors must be zero:
\[ \vec{A} \cdot (\vec{B} \times \vec{C}) = 0 \]
This leads us to the determinant condition:
\[ \begin{vmatrix} l_1 & m_1 & n_1 \\ l_2 & m_2 & n_2 \\ l_3 & m_3 & n_3 \end{vmatrix} = 0 \]

4. **Calculate the Determinant**:
Expanding the determinant, we have:
\[ l_1 (m_2 n_3 - m_3 n_2) - m_1 (l_2 n_3 - l_3 n_2) + n_1 (l_2 m_3 - l_3 m_2) = 0 \]
This implies that the vectors are linearly dependent.

5. **Relate Co-factors to Direction Cosines**:
The co-factors \( (p_1, q_1, r_1), (p_2, q_2, r_2), (p_3, q_3, r_3) \) can be interpreted as the direction cosines of the lines perpendicular to the original lines represented by A. Thus, we can conclude that:
\[ \vec{B} = k_1 \cdot \vec{A}, \quad \vec{C} = k_2 \cdot \vec{A} \]
for some scalars \( k_1 \) and \( k_2 \).

6. **Final Conclusion**:
Therefore, the vectors \( (p_1, q_1, r_1), (p_2, q_2, r_2), (p_3, q_3, r_3) \) represent the direction cosines of the lines perpendicular to the lines represented by \( (l_1, m_1, n_1), (l_2, m_2, n_2), (l_3, m_3, n_3) \).