Find the Determinant [[p,1,1],[q,1,q],[1,1,p]]

Problem

det(p)

Solution

1. Set up the determinant calculation using cofactor expansion along the first row.

2. Apply the formula for a 3×3 determinant by multiplying each element of the first row by the determinant of its corresponding 2×2 minor.

p*det(1)−1*det(q)+1*det(q)

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

p*(p−q)−1*(p*q−q)+1*(q−1)

4. Distribute the terms outside the parentheses.

p2−p*q−p*q+q+q−1

5. Combine like terms to simplify the expression.

p2−2*p*q+2*q−1

Final Answer

det(p)=p2−2*p*q+2*q−1

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