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

Problem

det(a2)

Solution

1. Apply the cofactor expansion along the first row to set up the determinant calculation.

det(a2)=a2*det(b)−a*det(b2)+1*det(b2)

2. Evaluate the 2×2 determinants using the formula det(e)=e*h−ƒ*g

a2*(b−c)−a*(b2−c2)+(b2*c−b*c2)

3. Factor out common terms from each part of the expression to simplify.

a2*(b−c)−a*(b−c)*(b+c)+b*c*(b−c)

4. Factor out the common binomial (b−c) from the entire expression.

(b−c)*(a2−a*(b+c)+b*c)

5. Expand and regroup the terms inside the second set of parentheses.

(b−c)*(a2−a*b−a*c+b*c)

6. Factor by grouping the terms a2−a*b−a*c+b*c as a*(a−b)−c*(a−b)

(b−c)*(a−c)*(a−b)

7. Rearrange the factors into the standard cyclic order (a−b)*(b−c)*(c−a) by factoring out a negative sign.

−(a−b)*(b−c)*(c−a)

Final Answer

det(a2)=−(a−b)*(b−c)*(c−a)

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