https://file.aiquickdraw.com/mathbot/file/17253866110503ihnqheu.jpg

To evaluate the determinant \(\Delta\) given in the image, we need to calculate the determinant of the 3x3 matrix:

\[
\Delta = \begin{vmatrix} 
a - b & m - n & x - y \\ 
b - c & n - p & y - z \\ 
c - a & p - m & z - x 
\end{vmatrix}
\]

To solve this, we'll use the cofactor expansion method along the first row. The determinant of a 3x3 matrix:

\[
\Delta = 
\begin{vmatrix}
a & b & c \\
d & e & f \\
g & h & i
\end{vmatrix}
\]
is given by:
\[
\Delta = a \begin{vmatrix} e & f \\ h & i \end{vmatrix} - b \begin{vmatrix} d & f \\ g & i \end{vmatrix} + c \begin{vmatrix} d & e \\ g & h \end{vmatrix}.
\]

Now, applying this formula to our matrix:

\[
\Delta = (a - b) 
\begin{vmatrix} 
n - p & y - z \\ 
p - m & z - x 
\end{vmatrix}
- (m - n) 
\begin{vmatrix} 
b - c & y - z \\ 
c - a & z - x 
\end{vmatrix}
+ (x - y) 
\begin{vmatrix} 
b - c & n - p \\ 
c - a & p - m 
\end{vmatrix}.
\]

Next, let's compute the 2x2 determinants one by one:

1. **First Minor:**
\[
\begin{vmatrix} 
n - p & y - z \\ 
p - m & z - x 
\end{vmatrix} 
= (n - p)(z - x) - (y - z)(p - m)
= nz - nx - pz + px - yp + zp + my - mz.
\]

2. **Second Minor:**
\[
\begin{vmatrix} 
b - c & y - z \\ 
c - a & z - x 
\end{vmatrix} 
= (b - c)(z - x) - (y - z)(c - a)
= bz - bx - cz + cx - yc + za + ay - az.
\]

3. **Third Minor:**
\[
\begin{vmatrix} 
b - c & n - p \\ 
c - a & p - m 
\end{vmatrix} 
= (b - c)(p - m) - (n - p)(c - a)
= bp - bm - cp + cm - nc + pc + an - ap.
\]

Substitute these values back into the determinant expression:

\[
\Delta = (a - b) [nz - nx - pz + px - yp + zp + my - mz] 
- (m - n) [bz - bx - cz + cx - yc + za + ay - az] 
+ (x - y) [bp - bm - cp + cm - nc + pc + an - ap].
\]

This expression represents the value of the determinant \(\Delta\). 

Would you like further details on any specific part, or is there something more you would like to explore regarding this determinant?

### Related Questions:
1. What are some properties of determinants that could simplify the calculation?
2. How can we use matrix row or column operations to simplify this determinant?
3. What are some real-world applications of determinants in linear algebra?
4. How can determinants be used to find the area of a triangle or parallelogram?
5. Can we interpret this determinant geometrically in some way?

**Tip:** When calculating determinants, look for opportunities to simplify the matrix using row or column operations, as these can often make the computation much easier!

Evaluate the determinant Δ given in the image.

Learn how to evaluate a 3x3 determinant using the cofactor expansion method. This problem involves key concepts in linear algebra and determinants, with detailed step-by-step calculations. Suitable for students in Grades 10-12.

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