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 `q.M_12 - y.M_22+ m.M_32` is :

Options

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

Answer

To solve the problem, we need to find the value of \( q \cdot M_{12} - y \cdot M_{22} + m \cdot M_{32} \) where \( M_{ij} \) denotes the minor 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 1: Calculate the minors

1. **Calculate \( M_{12} \)**:
- To find \( M_{12} \), we remove the first row and the second column from the determinant.
- The remaining matrix is:
\[ \begin{vmatrix} x & z \\ l & n \end{vmatrix} \]
- The minor \( M_{12} \) is calculated as:
\[ M_{12} = x \cdot n - z \cdot l \]

2. **Calculate \( M_{22} \)**:
- To find \( M_{22} \), we remove the second row and the second column from the determinant.
- The remaining matrix is:
\[ \begin{vmatrix} p & r \\ l & n \end{vmatrix} \]
- The minor \( M_{22} \) is calculated as:
\[ M_{22} = p \cdot n - r \cdot l \]

3. **Calculate \( M_{32} \)**:
- To find \( M_{32} \), we remove the third row and the second column from the determinant.
- The remaining matrix is:
\[ \begin{vmatrix} p & q \\ x & y \end{vmatrix} \]
- The minor \( M_{32} \) is calculated as:
\[ M_{32} = p \cdot y - q \cdot x \]

### Step 2: Substitute minors into the expression

Now we substitute the minors into the expression \( q \cdot M_{12} - y \cdot M_{22} + m \cdot M_{32} \):

\[ q \cdot M_{12} = q \cdot (x \cdot n - z \cdot l) = qxn - qzl \]

\[ -y \cdot M_{22} = -y \cdot (p \cdot n - r \cdot l) = -ypn + yrl \]

\[ m \cdot M_{32} = m \cdot (p \cdot y - q \cdot x) = mpy - mqx \]

### Step 3: Combine the results

Now we combine all these results:

\[ q \cdot M_{12} - y \cdot M_{22} + m \cdot M_{32} = (qxn - qzl) + (-ypn + yrl) + (mpy - mqx) \]

Combining like terms:

\[ = qxn - qzl - ypn + yrl + mpy - mqx \]

### Step 4: Rearranging the expression

Rearranging gives us:

\[ = (qxn - mqx) + (mpy - ypn) + (yrl - qzl) \]

### Final Result

The final expression simplifies to:

\[ = -\Delta \]

where \( \Delta \) is the determinant of the original matrix.

Thus, the value of \( q \cdot M_{12} - y \cdot M_{22} + m \cdot M_{32} \) is:

\[ \text{Final Answer: } -\Delta \]