Let P be a matrix of order 3 x 3 such that all the entries in P are from the set {-1, 0, 1}. Then, the maximum possible value of the determinant of P is ________.

det⁡(P)=a1   a2a3b1b2b3c1c2c3=a1b2c3+a2b3c1+a3b1c2−a3b2c1+a1b3c2+a2b1c3=x−ywhere x=a1b2c3+a2b3c1+a3b1c2 and y=a3b2c1+a1b3c2+a2b1c3Now, x≤3 and y≥−3.If we consider det.(P) = 6, thena1b2c3=a2b3c1=a3b1c2=1 and a3b2c1=a1b3c2=a2b1c3=−1, which is contradiction.So, next possibility is 5 for which one of the terms must be zero but that will make two terms zero.Hence, greatest value of determinant is 4.