Find the Determinant [[1,x,x^2],[1,x^2,x^4],[1,x^4,x^6]]

Problem

det(1)

Solution

1. Identify the matrix and the method for calculating the determinant of a 3×3 matrix. We will use cofactor expansion along the first column.

2. Expand along the first column by multiplying each element by its corresponding minor.

det(A)=1⋅det(x2)−1⋅det(x)+1⋅det(x)

3. Calculate the 2×2 determinants using the formula a*d−b*c

det(x2)=(x2)*(x6)−(x4)*(x4)=x8−x8=0

det(x)=(x)*(x6)−(x2)*(x4)=x7−x6

det(x)=(x)*(x4)−(x2)*(x2)=x5−x4

4. Combine the results into the expansion expression.

det(A)=0−(x7−x6)+(x5−x4)

5. Simplify the expression by distributing the negative sign and ordering the terms by degree.

det(A)=−x7+x6+x5−x4

Final Answer

det(1)=−x7+x6+x5−x4

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