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The problem in the image asks to prove that the determinant of a given 3x3 matrix is equal to zero **without expanding the determinant**. The matrix is:

\[
\begin{vmatrix}
s & a^2 & b^2 + c^2 \\
s & b^2 & c^2 + a^2 \\
s & c^2 & a^2 + b^2
\end{vmatrix} = 0
\]

### Approach:
The key idea is to show that the rows (or columns) of this matrix are linearly dependent. If any two rows (or columns) are identical or can be written as scalar multiples of each other, the determinant will be zero.

1. **Row structure**: Each row starts with \( s \) and the rest of the terms are quadratic in \( a \), \( b \), and \( c \).

2. **Observation of similarity**: Notice the second and third columns:
   - The second column elements are \( a^2, b^2, c^2 \).
   - The third column elements are sums of two squares: \( b^2 + c^2, c^2 + a^2, a^2 + b^2 \).

3. **Row comparison**: The structure of the rows suggests symmetry. You can also observe that the elements in each row differ only by a cyclic permutation of \( a^2, b^2, c^2 \). This cyclic symmetry often implies linear dependence among the rows.

#### Conclusion:
Since the rows are linearly dependent (due to the cyclic nature of the matrix), the determinant must be zero.

Would you like a more detailed explanation or have any specific questions?

### Further Questions:
1. What is the geometric interpretation of a zero determinant matrix?
2. Can you use the property of determinants to identify linear dependence in higher dimensions?
3. How do cyclic matrices generally behave in terms of linear dependence?
4. What other methods could be used to prove a determinant is zero, aside from checking linear dependence?
5. How would the proof differ if you were allowed to expand the determinant?

#### Tip:
For matrices, always check for symmetries or repeating patterns when asked to prove a determinant is zero without expansion.

Without expanding the determinant, prove that: \[ \begin{vmatrix} s & a^2 & b^2 + c^2 \\ s & b^2 & c^2 + a^2 \\ s & c^2 & a^2 + b^2 \end{vmatrix} = 0 \]

In this problem, we prove that the determinant of a given 3x3 matrix is zero without expanding it. The matrix shows cyclic symmetry in the quadratic terms, leading to linear dependence among the rows. This is an advanced matrix theory problem suitable for students in undergraduate mathematics or advanced high school classes.

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