Discussion

Question:

Suppose a, b and c are integers such that a > b > c > 0, and A = $[\begin{matrix} a & b & c \\ b & c & a \\ c & a & b \end{matrix}]$ . Then the value of the determinant of A

Explanation:

A = $[\begin{matrix} a & b & c \\ b & c & a \\ c & a & b \end{matrix}]$ .

Determinant of A = a(cb - a²) + b(ac - b²) + c(ba - c²)

= 3abc - a³ - b³ - c³

= - (a³ + b³ + c³ - 3abc)

= - (a + b + c)(a² + b² + c² - ab - bc - ca)

(a + b + c)(a² + b² + c² - ab - bc - ca) > 0 for a, b, c > 0

∴ Determinant of A = - (a + b + c)(a² + b² + c² - ab - bc - ca) is always negative.

Hence, option (c).

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