https://r2.erweima.ai/mathbot/image/172477461324584j9x95g.jpg

To solve the determinant in the image, let's first identify and analyze the given expression.

The determinant is:

\[
\begin{vmatrix}
1 + a + b & -2ab \\
\frac{2b}{1 - a + b} & \frac{2a}{1 - a - b}
\end{vmatrix}
\]

To solve the determinant, we use the standard formula for a 2x2 matrix determinant:

\[
\text{Determinant} = \text{(First element of the first row) * (Second element of the second row)} - \text{(Second element of the first row) * (First element of the second row)}
\]

Substituting the values from the determinant:

\[
\text{Determinant} = \left(1 + a + b\right) \times \frac{2a}{1 - a - b} - \left(-2ab\right) \times \frac{2b}{1 - a + b}
\]

Let's simplify this expression:

1. **Multiply the first part:**

\[
(1 + a + b) \times \frac{2a}{1 - a - b} = \frac{2a(1 + a + b)}{1 - a - b}
\]

2. **Multiply the second part (be careful with signs):**

\[
-(-2ab) \times \frac{2b}{1 - a + b} = \frac{4ab \cdot b}{1 - a + b} = \frac{4ab^2}{1 - a + b}
\]

Now combine the two terms:

\[
\text{Determinant} = \frac{2a(1 + a + b)}{1 - a - b} + \frac{4ab^2}{1 - a + b}
\]

This is the simplified expression for the determinant. Further simplification might be possible, but it depends on the context or specific algebraic manipulations. The determinant simplifies into the above expression based on the image provided.

Would you like to explore further details on this calculation or proceed with any other questions?

### Related Questions
1. What are the properties of determinants in 2x2 matrices?
2. How do you expand determinants for larger matrices?
3. Can we factorize the result obtained in this determinant?
4. How does the sign change affect determinant values?
5. What are the applications of determinants in solving linear equations?

### Tip
When solving determinants, always be cautious of sign changes, especially when dealing with expressions involving subtraction or negative numbers.

Learn how to solve the determinant of a 2x2 matrix with a detailed step-by-step guide. This problem involves understanding determinants, matrix operations, and algebraic manipulations. Suitable for undergraduate students studying linear algebra and matrix theory.

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