Find the Rank [[1,1,1],[a,b,c],[a^2,b^2,c^2]]=0

Problem

Rank*[[1,1,1],[a,b,c],[a2,b2,c2]]

Solution

1. Identify the matrix as a 3×3 Vandermonde matrix, denoted as V

2. Calculate the determinant of the matrix using the property of Vandermonde matrices.

det(V)=(b−a)*(c−a)*(c−b)

3. Analyze the condition for rank 3. The rank is 3 if and only if the determinant is non-zero.

det(V)≠0⇔a≠b,b≠c,a≠c

4. Analyze the condition for rank 2. If exactly two variables are equal (e.g., a=b≠c, the determinant is 0, but at least one 2×2 minor is non-zero.

|[1,1],[a,c]|=c−a≠0

5. Analyze the condition for rank 1. If all variables are equal (a=b=c, all rows become identical to the first row (or multiples of it if a=1.

[[1,1,1],[a,a,a],[a2,a2,a2]]

Final Answer

Rank*[[1,1,1],[a,b,c],[a2,b2,c2]]={[3,if *a,b,c* are distinct],[2,if exactly two are equal],[1,if *a=b=c])

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