Consider a $5\times5$ matrix $P=(5,4,3,2,1)$ which means it has anti-diagonal entries of $1$'s.
If we calculate $\det P$ using the theorem "The determinants changes sign when two rows are exchange", then it is : $$1 (making it identity)
But if we use cofactors then the answer is : $-1$
Is that possible? Which one is true?