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 `x.C_21+y.C_22+z.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 \( x \cdot C_{21} + y \cdot C_{22} + z \cdot C_{23} \) for the determinant \( \Delta = \begin{vmatrix} p & q & r \\ x & y & z \\ l & m & n \end{vmatrix} \).

### Step 1: Calculate the Determinant
We start by calculating the determinant \( \Delta \) using the first row:

\[ \Delta = p \begin{vmatrix} y & z \\ m & n \end{vmatrix} - q \begin{vmatrix} x & z \\ l & n \end{vmatrix} + r \begin{vmatrix} x & y \\ l & m \end{vmatrix} \]

Calculating the 2x2 determinants:

1. \( \begin{vmatrix} y & z \\ m & n \end{vmatrix} = yn - mz \)
2. \( \begin{vmatrix} x & z \\ l & n \end{vmatrix} = xn - lz \)
3. \( \begin{vmatrix} x & y \\ l & m \end{vmatrix} = xm - ly \)

So, substituting these back into the determinant:

\[ \Delta = p(yn - mz) - q(xn - lz) + r(xm - ly) \]

### Step 2: Calculate the Cofactors
Next, we need to find the cofactors \( C_{21} \), \( C_{22} \), and \( C_{23} \).

The cofactor \( C_{ij} \) is given by:

\[ C_{ij} = (-1)^{i+j} M_{ij} \]

where \( M_{ij} \) is the minor of the element in the \( i^{th} \) row and \( j^{th} \) column.

1. **Cofactor \( C_{21} \)**:
- Minor \( M_{21} \) is obtained by deleting the 2nd row and 1st column:
\[ M_{21} = \begin{vmatrix} q & r \\ m & n \end{vmatrix} = qn - rm \]
- Thus, \( C_{21} = (-1)^{2+1} M_{21} = -(qn - rm) = rm - qn \).

2. **Cofactor \( C_{22} \)**:
- Minor \( M_{22} \) is obtained by deleting the 2nd row and 2nd column:
\[ M_{22} = \begin{vmatrix} p & r \\ l & n \end{vmatrix} = pn - lr \]
- Thus, \( C_{22} = (-1)^{2+2} M_{22} = pn - lr \).

3. **Cofactor \( C_{23} \)**:
- Minor \( M_{23} \) is obtained by deleting the 2nd row and 3rd column:
\[ M_{23} = \begin{vmatrix} p & q \\ l & m \end{vmatrix} = pm - ql \]
- Thus, \( C_{23} = (-1)^{2+3} M_{23} = -(pm - ql) = ql - pm \).

### Step 3: Substitute Cofactors into the Expression
Now we substitute the cofactors back into the expression \( x \cdot C_{21} + y \cdot C_{22} + z \cdot C_{23} \):

\[ x \cdot C_{21} + y \cdot C_{22} + z \cdot C_{23} = x(rm - qn) + y(pn - lr) + z(ql - pm) \]

Expanding this:

\[ = xrm - xqn + ypn - ylr + zql - zpm \]

### Step 4: Combine Like Terms
Rearranging the terms gives:

\[ = (xrm - zpm) + (ypn - ylr) + (zql - xqn) \]

### Final Result
Thus, we can express the final result as:

\[ = \Delta \]

where \( \Delta \) is the determinant we calculated initially.

### Conclusion
The value of \( x \cdot C_{21} + y \cdot C_{22} + z \cdot C_{23} \) is equal to the determinant \( \Delta \).