Consider the determinant, `Delta=|(p,q,r),(x,y,z),(l,m,n)|` .
`M_(ij)` denotes the minor of an element in `i^(th)` row, and `j^(th)` column
`C_(ij)` denotes the cofactor of an element in `i^(th)` row and `j^(th)` column
The value of `p.C_21+q.C_22+r.C_23` is :

Options

A. 0
B. `-Delta`
C. `Delta`
D. `Delta^2`

Answer

To solve the problem, we need to find the value of \( p \cdot C_{21} + q \cdot C_{22} + r \cdot C_{23} \) where \( C_{ij} \) denotes the cofactor of the element in the \( i^{th} \) row and \( j^{th} \) column of the determinant \( \Delta = \begin{vmatrix} p & q & r \\ x & y & z \\ l & m & n \end{vmatrix} \).

### Step-by-Step Solution:

1. **Calculate the minors \( M_{21}, M_{22}, M_{23} \)**:
- The minor \( M_{21} \) is obtained by deleting the 2nd row and 1st column:
\[ M_{21} = \begin{vmatrix} q & r \\ m & n \end{vmatrix} = qn - rm \]
- The minor \( M_{22} \) is obtained by deleting the 2nd row and 2nd column:
\[ M_{22} = \begin{vmatrix} p & r \\ l & n \end{vmatrix} = pn - rl \]
- The minor \( M_{23} \) is obtained by deleting the 2nd row and 3rd column:
\[ M_{23} = \begin{vmatrix} p & q \\ l & m \end{vmatrix} = pm - ql \]

2. **Calculate the cofactors \( C_{21}, C_{22}, C_{23} \)**:
- The cofactor \( C_{21} \) is given by:
\[ C_{21} = (-1)^{2+1} M_{21} = -M_{21} = -(qn - rm) \]
- The cofactor \( C_{22} \) is given by:
\[ C_{22} = (-1)^{2+2} M_{22} = M_{22} = pn - rl \]
- The cofactor \( C_{23} \) is given by:
\[ C_{23} = (-1)^{2+3} M_{23} = -M_{23} = -(pm - ql) \]

3. **Substitute the cofactors into the expression \( p \cdot C_{21} + q \cdot C_{22} + r \cdot C_{23} \)**:
\[ p \cdot C_{21} = p \cdot (-(qn - rm)) = -pqn + prm \]
\[ q \cdot C_{22} = q \cdot (pn - rl) = qpn - qrl \]
\[ r \cdot C_{23} = r \cdot (-(pm - ql)) = -rpm + rql \]

4. **Combine all the terms**:
\[ p \cdot C_{21} + q \cdot C_{22} + r \cdot C_{23} = (-pqn + prm) + (qpn - qrl) + (-rpm + rql) \]
\[ = -pqn + qpn + prm - qrl - rpm + rql \]
\[ = (q - p)pn + (r - q)ql + (p - r)rm \]

5. **Recognizing the expression**:
The expression simplifies to zero because it represents a linear combination of the rows of the determinant, which implies that the determinant is zero.

### Final Result:
\[ p \cdot C_{21} + q \cdot C_{22} + r \cdot C_{23} = 0 \]