If `A=|(1,1,1),(a,b,c),(a^3,b^3,c^3)|, B=|(1,1,1),(a^2,b^2,c^2),(a^3,b^3,c^3)|, C=|(a,b,c),(a^2,b^2,c^2),(a^3,b^3,c^3)|` , then which relation is correct :

Options

A. A=B
B. A=C
C. B=C
D. None of these

Answer

To solve the problem, we need to find the determinants of the matrices \( A \), \( B \), and \( C \) given by:

\[ A = \begin{vmatrix} 1 & 1 & 1 \\ a & b & c \\ a^3 & b^3 & c^3 \end{vmatrix}, \quad B = \begin{vmatrix} 1 & 1 & 1 \\ a^2 & b^2 & c^2 \\ a^3 & b^3 & c^3 \end{vmatrix}, \quad C = \begin{vmatrix} a & b & c \\ a^2 & b^2 & c^2 \\ a^3 & b^3 & c^3 \end{vmatrix} \]

### Step 1: Calculate the Determinant of Matrix \( A \)

Using the properties of determinants, we can expand the determinant \( A \):

\[ A = 1 \cdot \begin{vmatrix} b & c \\ b^3 & c^3 \end{vmatrix} - 1 \cdot \begin{vmatrix} a & c \\ a^3 & c^3 \end{vmatrix} + 1 \cdot \begin{vmatrix} a & b \\ a^3 & b^3 \end{vmatrix} \]

Calculating these 2x2 determinants:

1. \( \begin{vmatrix} b & c \\ b^3 & c^3 \end{vmatrix} = bc^3 - b^3c = b(c^3 - b^2) \)
2. \( \begin{vmatrix} a & c \\ a^3 & c^3 \end{vmatrix} = ac^3 - a^3c = a(c^3 - a^2) \)
3. \( \begin{vmatrix} a & b \\ a^3 & b^3 \end{vmatrix} = ab^3 - a^3b = a(b^3 - a^2) \)

Thus, we have:

\[ A = b(c^3 - b^2) - a(c^3 - a^2) + a(b^3 - a^2) \]

### Step 2: Calculate the Determinant of Matrix \( B \)

Similarly, we can calculate the determinant \( B \):

\[ B = 1 \cdot \begin{vmatrix} b^2 & c^2 \\ b^3 & c^3 \end{vmatrix} - 1 \cdot \begin{vmatrix} a^2 & c^2 \\ a^3 & c^3 \end{vmatrix} + 1 \cdot \begin{vmatrix} a^2 & b^2 \\ a^3 & b^3 \end{vmatrix} \]

Calculating these 2x2 determinants:

1. \( \begin{vmatrix} b^2 & c^2 \\ b^3 & c^3 \end{vmatrix} = b^2c^3 - b^3c^2 = b^2(c^3 - bc) \)
2. \( \begin{vmatrix} a^2 & c^2 \\ a^3 & c^3 \end{vmatrix} = a^2c^3 - a^3c^2 = a^2(c^3 - ac) \)
3. \( \begin{vmatrix} a^2 & b^2 \\ a^3 & b^3 \end{vmatrix} = a^2b^3 - a^3b^2 = a^2(b^3 - ab) \)

Thus, we have:

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

### Step 3: Calculate the Determinant of Matrix \( C \)

Now, we calculate the determinant \( C \):

\[ C = \begin{vmatrix} a & b & c \\ a^2 & b^2 & c^2 \\ a^3 & b^3 & c^3 \end{vmatrix} \]

This is a Vandermonde determinant, which can be calculated as:

\[ C = (b-a)(c-a)(c-b) \]

### Step 4: Compare the Determinants

Now we need to compare the determinants \( A \), \( B \), and \( C \):

- \( A \) and \( B \) are not equal because they have different structures and terms.
- \( A \) and \( C \) are also not equal since \( C \) is a product of differences.
- \( B \) and \( C \) are not equal for the same reasons.

### Conclusion

Since none of the matrices \( A \), \( B \), or \( C \) are equal to each other, the correct relation is:

**None of these.**