Step 1: Setting up the matrix for determinant calculation.
We are given the matrix:
\[
M = \begin{bmatrix}
\frac{a^2 + b^2}{c} & c & c
a & \frac{b^2 + c^2}{a} & a
b & b & \frac{c^2 + a^2}{b}
\end{bmatrix}
\]
Step 2: Calculate the determinant.
To find the determinant of the 3x3 matrix \( M \), we use the formula for the determinant of a 3x3 matrix:
\[
\text{det}(M) = m_{11}(m_{22}m_{33} - m_{23}m_{32}) - m_{12}(m_{21}m_{33} - m_{23}m_{31}) + m_{13}(m_{21}m_{32} - m_{22}m_{31})
\]
where \( m_{ij} \) represents the element in the \( i \)-th row and \( j \)-th column of matrix \( M \).
Substituting the elements of matrix \( M \):
\[
\text{det}(M) = \left(\frac{a^2 + b^2}{c}\right)\left(\frac{b^2 + c^2}{a}\frac{c^2 + a^2}{b} - a \cdot b\right) - c\left(a \cdot \frac{c^2 + a^2}{b} - a \cdot b\right) + c\left(a \cdot b - b \cdot \frac{b^2 + c^2}{a}\right)
\]
Step 3: Simplification and result.
Let's simplify each term step by step.
1. First term:
\[
\left(\frac{a^2 + b^2}{c}\right)\left(\frac{b^2 + c^2}{a}\frac{c^2 + a^2}{b} - a \cdot b\right)
\]
Simplify the product inside the parentheses:
\[
\frac{b^2 + c^2}{a} \cdot \frac{c^2 + a^2}{b} = \frac{(b^2 + c^2)(c^2 + a^2)}{ab}
\]
Thus, the first term becomes:
\[
\frac{a^2 + b^2}{c} \left(\frac{(b^2 + c^2)(c^2 + a^2)}{ab} - ab\right)
\]
2. Second term:
\[
- c\left(a \cdot \frac{c^2 + a^2}{b} - a \cdot b\right)
\]
Simplify inside the parentheses:
\[
a \cdot \frac{c^2 + a^2}{b} - a \cdot b = \frac{a(c^2 + a^2)}{b} - ab = \frac{a(c^2 + a^2) - a b^2}{b} = \frac{a(c^2 + a^2 - b^2)}{b}
\]
Thus, the second term becomes:
\[
- c \cdot \frac{a(c^2 + a^2 - b^2)}{b} = -\frac{a c (c^2 + a^2 - b^2)}{b}
\]
3. Third term:
\[
c\left(a \cdot b - b \cdot \frac{b^2 + c^2}{a}\right)
\]
Simplify inside the parentheses:
\[
a \cdot b - b \cdot \frac{b^2 + c^2}{a} = ab - \frac{b(b^2 + c^2)}{a} = \frac{a^2 b - b(b^2 + c^2)}{a} = \frac{b(a^2 - b^2 - c^2)}{a}
\]
Thus, the third term becomes:
\[
c \cdot \frac{b(a^2 - b^2 - c^2)}{a} = \frac{b c (a^2 - b^2 - c^2)}{a}
\]
Combining all terms:
\[
\text{det}(M) = \frac{a^2 + b^2}{c} \left(\frac{(b^2 + c^2)(c^2 + a^2)}{ab} - ab\right) - \frac{a c (c^2 + a^2 - b^2)}{b} + \frac{b c (a^2 - b^2 - c^2)}{a}
\]
After further algebraic simplification and recognizing symmetries, the determinant simplifies to:
\[
\text{det}(M) = 4abc
\]
Final Answer:
\[
\boxed{4abc}
\]
