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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 $\left| {\,\begin{array}{*{20}{c}}a&b&c\\b&c&a\\c&a&b\end{array}\,} \right| = k(a + b + c)({a^2} + {b^2} + {c^2}$ $ - bc - ca - ab)$, then $k =$
  • A
    $1$
  • B
    $2$
  • ✓
    $-1$
  • D
    $-2$
✓

Answer

Correct option: C.
$-1$
c
(c) $\left| {\,\begin{array}{*{20}{c}}a&b&c\\b&c&a\\c&a&b\end{array}\,} \right| = a(bc - {a^2}) - b({b^2} - ca) + c(ab - {c^2})$

= $ - {a^3} - {b^3} - {c^3} + 3abc$ = $ - 1\,[{a^3} + {b^3} + {c^3} - 3abc]$

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

==> $k = - 1$.

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