Advent calendar 2022
12 December
The determinant of the 2 by 2 matrix \(\begin{pmatrix}a&b\\c&d\end{pmatrix}\) is \(ad-bc\).
If a 2 by 2 matrix's entries are all in the set \(\{1, 2, 3\}\), the largest
possible deteminant of this matrix is 8.
What is the largest possible determinant of a 2 by 2 matrix whose entries are all in the set
\(\{1, 2, 3, ..., 12\}\)?
Show answer & extension
Hide answer & extension
The largest deteminant will be made by making \(a\) and \(d\) as large as possible (ie 12) and \(b\) and \(c) as small as possible (ie 1). This gives a
determinant of 143.
Extension
The determinant of the 3 by 3 matrix \(\begin{pmatrix}a&b&c\\d&e&f\\g&h&i\end{pmatrix}\) is \(a(ei-fh)-b(di-fg)+c(dh-eg)\).
If a 3 by 3 matrix's entries are all in the set \(\{1, 2, 3\}\), the largest possible deteminant of this matrix is 28.
What is the largest possible determinant of a 3 by 3 matrix whose entries are all in the set
\(\{1, 2, 3, ..., 12\}\)?
