Find the Determinant [[x,y,z],[x^2,y^2,z^2],[x^3,y^3,z^3]]

Problem

det(x)

Solution

1. Factor out common terms from each column. We can factor x from the first column, y from the second column, and z from the third column.

x*y*z*det(1)

2. Identify the remaining matrix as a Vandermonde matrix of order 3. The determinant of a Vandermonde matrix det(V(x,y,z)) is given by the product of the differences of its variables.

det(1)=(y−x)*(z−x)*(z−y)

3. Combine the factored terms with the Vandermonde determinant result to find the final expression.

x*y*z*(y−x)*(z−x)*(z−y)

Final Answer

det(x)=x*y*z*(y−x)*(z−x)*(z−y)

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