JEE Advanced 2018 Paper 2Question 1
Let $P$ be a matrix of order $3\times3$ 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 _____ .
Determinant
Answer$4$
Solution
Maximum $\det$ with entries in $\{-1,0,1\}$ is $4$, e.g. $\begin{vmatrix}1&1&1\\1&-1&1\\1&1&-1\end{vmatrix}=4$ (Hadamard bound gives at most $3\sqrt3<5.2$, and the determinant is an integer that is even for such matrices). So the maximum is $4$.