If a,b,c be positive and not all equal, then value of determinant left| {,begin{array}{*{20}{c}}a&am

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3 and 4 .Determinants and Matrices

medium

If $a,b,c$ be positive and not all equal, then the value of the determinant $\left| {\,\begin{array}{*{20}{c}}a&b&c\\b&c&a\\c&a&b\end{array}\,} \right|$ is

A

$- ve$

B

$=+ ve$

C

Depends on $a,b,c$

D

None of these

(IIT-1982)

Solution

(a) $\Delta = – ({a^3} + {b^3} + {c^3} – 3abc)$

= $ – (a + b + c)\,({a^2} + {b^2} + {c^2} – ab – bc – ca)$

$ = – \frac{1}{2}(a + b + c)\,[{(a – b)^2} + {(b – c)^2} + {(c – a)^2}]$,

which is clearly negative because of the given conditions.

Standard 12

Mathematics

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