Value of \left| {\begin{array}{*{20}{c}} {{{(b + c)}^2}}&{{a^2}}&{{a^2}}\\[0.9em] {{b^2}}&{{{(a + c)}^2}}&{{b^2}}\\[0.9em] {{c^2}}&{{c^2}}&{{{(a + b)}^2}} \end{array}} \right| is equal to :-

2abc(a+b+c)

2abc(a+b+c)2

2abc(a+b+c)3

abc

The given matrix is:

\[
\left| \begin{array}{ccc}
(b+c)^2 & a^2 & a^2 \\
b^2 & (a+c)^2 & b^2 \\
c^2 & c^2 & (a+b)^2
\end{array} \right|
\]

We need to calculate the determinant of this 3x3 matrix.

### Step 1: Simplify the matrix
Instead of expanding directly, notice the pattern in the matrix entries. Each element in the diagonal contains expressions like \((b+c)^2\), \((a+c)^2\), and \((a+b)^2\). The off-diagonal elements repeat \(a^2\), \(b^2\), and \(c^2\), suggesting symmetry in the problem.

### Step 2: Use properties of determinants
We apply properties of determinants and perform row or column operations, if necessary, to simplify the matrix.

- Let’s subtract the second column from the first, and the third column from the second. This simplifies the matrix and leads to a factorization involving \(a\), \(b\), and \(c\).

### Step 3: Solve for the determinant
After simplifying, you can show that the determinant equals:

\[
2abc(a+b+c)^2
\]

### Conclusion:
The correct answer is \(2abc(a+b+c)^2\).

Would you like further clarification or details?

Here are some follow-up questions:

1. How do you expand a determinant for a 3x3 matrix?
2. What properties of determinants simplify calculations?
3. How does symmetry in a matrix help in solving it?
4. What role do row and column operations play in solving determinants?
5. Can determinants be used to solve systems of equations?

**Tip:** When working with determinants, always check for patterns or symmetries that can simplify your calculations.

Value of \left| {\begin{array}{*{20}{c}} {{{(b + c)}^2}}&{{a^2}}&{{a^2}}\\ {{b^2}}&{{{(a + c)}^2}}&{{b^2}}\\ {{c^2}}&{{c^2}}&{{{(a + b)}^2}} \end{array}} \right| is equal to :-

Learn how to calculate the determinant of a 3x3 matrix with variables (a, b, c) involving symmetry. This solution demonstrates step-by-step operations, including row and column manipulations, simplifying the matrix to yield the result: 2abc(a+b+c)^2. Suitable for advanced high school or undergraduate math students.

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