Suppose \(a, b\) and \(c\) are integers such that \(a>b>c>0\), and \(A=\left[\begin{array}{lll}a & b & c \\ b & c & a \\ c & a & b\end{array}\right]\). Then the determinant of \(A\)

Question and options

Suppose

a, b

and

c

are integers such that

a>b>c>0

, and

A=\left[\begin{array}{lll}a & b & c \\ b & c & a \\ c & a & b\end{array}\right]

. Then the determinant of

A

Correct answer: C - is negative

det A = 3abc-a^3-b^3-c^3 = -(a+b+c)(a^2+b^2+c^2-ab-bc-ca) = -(1/2)(a+b+c)[(a-b)^2+(b-c)^2+(c-a)^2] < 0 since a,b,c are distinct positive integers.