Divisibility of the Determinant of a Class of Matrices ... - MAA Sections

dimension. First, we look at examples involving 2-by-2 matrices to see if this
case holds: if we are given two two-digit integers (say, x = a 1 a 2 and y = b 1 b 2 )
and if these integers are divisible by a given integer k, is it true that k also
a1 a
divides the determinant of the matrix
2
b 1 b 2
?
We investigate this case and see if it extends to higher dimensions, that
is to the 3-by-3 case, to the 4-by-4 case, and in general, to the n-by-n case.
Although the original problem gave a very specific matrix, we solve a generalization
of this problem and also look at other problems regarding the
determinant of the matrix.
We let Z be the set of all integers, we let N be the set of all natural
numbers, and we let M n (Z) be the set of all n-by-n matrix with integer
entries. We also denote the determinant of a matrix A by det(A).
A two digit natural number ab is really a(10)+b, and a three digit natural
number abc is actually a(10 2 )+b(10 1 )+c(10 0 ). In general, an n digit natural
number a 1 a 2 a 3 a 4 · · · a n actually means
a 1 10 n−1 + a 2 (10 n−2 ) + a 3 (10 n−3 ) + · · · + a n−1 (10 1 ) + a n (10 0 )
where 0 ≤ a i ≤ 9 for each i = 1, 2, ..., n, and a 1 ≠ 0.
Example 1 The numbers 72 and 18 are divisible by 3. The matrix
7 2
1 8
has determinant 7(8) − 2(1) = 54, which is also divisible by 3.
Example 2 The numbers 16 and 36 are divisible by 4. The matrix
1 6
3 6
has determinant 1(6) − 6(3) = −12, which is divisible by 4 as well.
In these examples, we see that if two two-digit positive integers a 1 a 2 and
b 1 b 2 are multiples of a positive integer k, then the determinant of the matrix
a1 a
A =
2
(which is a
b 1 b 1 b 2 − a 2 b 1 ), is also divisible by k. Note that even
2
when the determinant of A is 0, then k also divides the determinant of A.
2