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3 and 4 . determinant and metrices — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths3 and 4 . determinant and metrices1 Mark
MCQ
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
  • ✓
    $- ve$
  • B
    $=+ ve$
  • C
    Depends on $a,b,c$
  • D
    None of these
✓

Answer

Correct option: A.
$- ve$
a
(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.

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