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STD 12 - 3 and 4 . determinant and metrices — Mathematics STD 12 Science — Question

CBSE BoardEnglish MediumSTD 12 ScienceMathematicsSTD 12 - 3 and 4 . determinant and metrices1 Mark
MCQ
The value of $\left| {\,\begin{array}{*{20}{c}}1&1&1\\{bc}&{ca}&{ab}\\{b + c}&{c + a}&{a + b}\end{array}\,} \right|$is
  • A
    $1$
  • B
    $0$ 
  • ✓
    $(a - b)(b - c)(c - a)$
  • D
    $(a + b)(b + c)(c + a)$
✓

Answer

Correct option: C.
$(a - b)(b - c)(c - a)$
c
 (c) $\left| {\,\begin{array}{*{20}{c}}1&1&1\\{bc}&{ca}&{ab}\\{b + c}&{c + a}&{a + b}\end{array}\,} \right|$=$\left| {\,\begin{array}{*{20}{c}}0&0&1\\{c(b - a)}&{a(c - b)}&{ab}\\{b - a}&{c + a}&{a + b}\end{array}\,} \right|$

$\{ {C_1} \to {C_1} - {C_2},\,{C_2} \to {C_2} - {C_3}\} $

= $(b - a)\,\,(c - b)\,\,\left| {\,\begin{array}{*{20}{c}}0&0&1\\c&a&{ab}\\1&1&{a + b}\end{array}\,} \right|$= $(b - a)\,(c - a)\,\,(c - a)$

$ = (a - b)\,\,(b - c)\,\,(c - a)$.

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