Maximum Possible Value of Determinant of a 3x3 Matrix

To find the maximum possible value of the determinant of a 3x3 matrix P, we need to consider all the possible combinations of entries from the set {-1, 0, 1}.

Enumerating All Possible Matrices

We can enumerate all the possible matrices of order 3x3 by considering each entry one by one. Since there are 9 entries in a 3x3 matrix, each entry can take one of three values: -1, 0, or 1. Therefore, the total number of possible matrices is 3^9 = 19683.

Analyzing the Determinant

The determinant of a 3x3 matrix can be calculated as follows:

det(P) = a(ei - fh) - b(di - fg) + c(dh - eg)

where P = [a b c; d e f; g h i].

Key Observation

The key observation is that the terms in the determinant formula are either positive or negative. The maximum possible value of the determinant will be achieved when we choose the entries to maximize the positive terms and minimize the negative terms.

Maximizing Positive Terms

To maximize the positive terms, we need to choose entries such that the products in the positive terms are as large as possible. Since the entries can only be -1, 0, or 1, we want to choose entries that are 1 in the positive terms.

Minimizing Negative Terms

To minimize the negative terms, we need to choose entries such that the products in the negative terms are as small as possible. Since the entries can only be -1, 0, or 1, we want to choose entries that are -1 in the negative terms.

Optimal Matrix

Based on the above observations, the optimal matrix that maximizes the determinant can be constructed as follows:

P = [1 1 1; 1 1 1; 1 1 1]

Determinant of Optimal Matrix

Calculating the determinant of the optimal matrix:

det(P) = 1(1*1 - 1*1) - 1(1*1 - 1*1) + 1(1*1 - 1*1)
= 1(0) - 1(0) + 1(0)
= 0

Therefore, the maximum possible value of the determinant of a 3x3 matrix P, where all the entries are from the set {-1, 0, 1}, is 0.