Find the Determinant [[a^4,b^4,c^4],[a^2,b^2,c^2],[1,1,1]]

Problem

det(a4)

Solution

1. Identify the matrix as a Vandermonde-style matrix where the variables are a2 b2 and c2

2. Perform row operations to create zeros in the third row by subtracting the first column from the second and third columns.

(C_2)→(C_2)−(C_1)

(C_3)→(C_3)−(C_1)

3. Rewrite the determinant after the column operations.

det(a4)

4. Expand along the third row to reduce the matrix to a 2×2 determinant.

1⋅det(b4−a4)

5. Factor the terms in the first row using the difference of squares formula x2−y2=(x−y)*(x+y)

det((b2−a2)*(b2+a2))

6. Factor out common terms (b2−a2) from the first column and (c2−a2) from the second column.

(b2−a2)*(c2−a2)*det(b2+a2)

7. Calculate the 2×2 determinant.

(b2−a2)*(c2−a2)*((b2+a2)*(1)−(c2+a2)*(1))

8. Simplify the expression inside the parentheses.

(b2−a2)*(c2−a2)*(b2+a2−c2−a2)

(b2−a2)*(c2−a2)*(b2−c2)

9. Rearrange the factors to follow the standard cyclic order (a2−b2)*(b2−c2)*(c2−a2) by factoring out a negative sign.

−(a2−b2)*(b2−c2)*(c2−a2)

Final Answer

det(a4)=−(a2−b2)*(b2−c2)*(c2−a2)

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