Can you do the actual calculation and solve 

from sympy import Matrix, symbols, simplify

# Define symbols
a, b, c = symbols('a b c')

# Define the matrix A
A = Matrix([
    [b + c, a, b*c],
    [b, c + a, b],
    [c, c, a + b]
])

# Calculate the determinant of matrix A
det_A = A.det()

# Simplify the determinant expression
simplified_det_A = simplify(det_A)
simplified_det_AThe determinant of the matrix \( A \) simplifies to:

\[
\Delta = c(a^2 - abc + 3ab + ac + b^2c - bc^2)
\]

This expression isn't in the exact form of \( 4abc \), but let's break it down further by substituting specific values for \( a \), \( b \), and \( c \), or looking for potential factorizations that could help simplify it further.

Would you like to proceed with factoring or substituting specific values for further simplification?

Learn how to calculate and simplify the determinant of a 3x3 matrix with step-by-step instructions. This problem involves matrix algebra and determinants, covering the determinant formula for 3x3 matrices. Suitable for advanced mathematics enthusiasts.

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