`|A-B| ne 0`, `A^(4)=B^(4)`, `C^(3)A=C^(3)B`, `B^(3)A=A^(3)B`, then `|A^(3)+B^(3)+C^(3)|=`

Options

A. `0`
B. `1`
C. `3|A|^(3)`
D. `6`

Explanation

To solve the problem step by step, we will analyze the given conditions and derive the required determinant.

### Step 1: Understand the Given Conditions
We are given:
1. \(|A - B| \neq 0\) (the determinant of \(A - B\) is not equal to zero)
2. \(A^4 = B^4\)
3. \(C^3 A = C^3 B\)
4. \(B^3 A = A^3 B\)

### Step 2: Use the Identity for Cubes
We know the identity for the sum of cubes:
\[ A^3 + B^3 = (A + B)(A^2 - AB + B^2) \]
Thus, we can express \(A^3 + B^3 + C^3\) as:
\[ A^3 + B^3 + C^3 = (A + B)(A^2 - AB + B^2) + C^3 \]

### Step 3: Factor Out \(A - B\)
We can also use the identity for the difference of cubes:
\[ A^3 - B^3 = (A - B)(A^2 + AB + B^2) \]
This means we can express \(A^3 + B^3\) using \(A - B\):
\[ A^3 + B^3 = (A - B)(A^2 + AB + B^2) + 2B^3 \]

### Step 4: Combine the Terms
Now, we can write:
\[ A^3 + B^3 + C^3 = (A - B)(A^2 + AB + B^2) + C^3 \]

### Step 5: Apply the Given Conditions
From the condition \(A^4 = B^4\), we can derive that:
\[ A^4 - B^4 = (A^2 - B^2)(A^2 + B^2) = 0 \]
This implies \(A^2 = B^2\) or \(A^2 + B^2 = 0\). Since \(|A - B| \neq 0\), we can conclude that \(A^2 = B^2\).

### Step 6: Evaluate the Determinant
Now, we can evaluate the determinant:
\[ |A^3 + B^3 + C^3| = |(A - B)(A^2 + AB + B^2) + C^3| \]
Since \(|A - B| \neq 0\), we can conclude that the determinant of the entire expression must equal zero.

### Final Result
Thus, we have:
\[ |A^3 + B^3 + C^3| = 0 \]