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Let's go through the solutions for both Example 1 and Example 2.

### Example 1:
You are asked to show that the determinant of the following matrix vanishes (i.e., equals zero) without expanding:
\[
\Delta = \begin{vmatrix}
1 & x & y + z \\
1 & y & z + x \\
1 & z & x + y
\end{vmatrix}
\]

#### Solution:
1. **Apply a column operation:** Add the first column (\(C_1\)) to the third column (\(C_3\)):
   \[
   \Delta = \begin{vmatrix}
   1 & x & y + z \\
   1 & y & z + x \\
   1 & z & x + y
   \end{vmatrix}
   \]
   becomes
   \[
   \Delta = \begin{vmatrix}
   1 & x & x + y + z \\
   1 & y & x + y + z \\
   1 & z & x + y + z
   \end{vmatrix}
   \]

2. **Factor out** \((x + y + z)\) from the third column (\(C_3\)):
   \[
   \Delta = (x + y + z) \begin{vmatrix}
   1 & x & 1 \\
   1 & y & 1 \\
   1 & z & 1
   \end{vmatrix}
   \]

3. **Observe the determinant structure:** The determinant of this matrix is 0 because the first and third columns are identical:
   \[
   \begin{vmatrix}
   1 & x & 1 \\
   1 & y & 1 \\
   1 & z & 1
   \end{vmatrix} = 0
   \]

Thus, \(\Delta = (x + y + z) \times 0 = 0\), which shows that the determinant vanishes.

---

### Example 2:
You are asked to show that the determinant of the following matrix equals zero without expanding:
\[
|A| = \begin{vmatrix}
1 & \omega & \omega^2 \\
\omega & \omega^2 & 1 \\
\omega^2 & 1 & \omega
\end{vmatrix} = 0
\]

Here, \(\omega\) is a cube root of unity, meaning that \(\omega^3 = 1\) and \(\omega^2 + \omega + 1 = 0\).

#### Solution:
1. **Use the properties of cube roots of unity:** We know the relations:
   \[
   \omega^3 = 1 \quad \text{and} \quad 1 + \omega + \omega^2 = 0
   \]

2. **Apply this to simplify the determinant:** Notice that each row of the matrix is a cyclic permutation of \((1, \omega, \omega^2)\). This kind of matrix, where rows (or columns) are cyclic permutations of each other, has a determinant of 0, as shown by standard results in determinant theory.

Thus, without needing to fully expand, you can conclude that \(|A| = 0\).

---

Let me know if you'd like further clarification!

### Relative Questions:
1. Why does adding columns or rows of a matrix not change the determinant in certain cases?
2. How does the property of cyclic matrices help in reducing determinant calculations?
3. What are cube roots of unity, and how are they used in determinant properties?
4. Can you explain why identical rows or columns lead to a determinant of 0?
5. How does factorizing a common term from a column affect the determinant?

### Tip:
When solving determinant problems, always check for symmetry or common factors across rows or columns to simplify your calculations quickly.

Solve example 1 and example 2 without expanding the determinant.

Learn how to solve determinant problems using matrix operations and cube roots of unity. Detailed step-by-step solutions for proving that certain 3x3 matrix determinants vanish without expanding.

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