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  • The value of determinant A. a3 + b3 + c3

The value of determinant \left|\begin{array}{lll} a-b & b+c & a \\ b-c & c+a & b \\ c-a & a+b & c \end{array}\right|

A. a^3 + b^3 + c^3

B. 3 bc

C. a^3 + b^3 + c^3-3abc

D. none of these}

Answers (1)

C)

Given:

\left|\begin{array}{lll} a-b & b+c & a \\ b-c & c+a & b \\ c-a & a+b & c \end{array}\right|

Apply C2→ C2 + C3

\\\begin{aligned} &=\left|\begin{array}{lll} a-b & a+b+c & a \\ b-c & c+a+b & b \\ c-a & a+b+c & c \end{array}\right|\\ &\text { Take }(a+b+c) \text { common from Column } 2\\ &=(a+b+c)\left|\begin{array}{lll} a-b & 1 & a \\ b-c & 1 & b \\ c-a & 1 & c \end{array}\right|\\ &\text { Apply } \mathrm{C}_{1} \rightarrow \mathrm{C}_{1}-\mathrm{C}_{3}\\ &=(a+b+c)\left|\begin{array}{lll} a-b-a & 1 & a \\ b-c-b & 1 & b \\ c-a-c & 1 & c \end{array}\right|\\ &=(a+b+c)\left|\begin{array}{ccc} -b & 1 & a \\ -c & 1 & b \\ -a & 1 & c \end{array}\right| \end{aligned}

Expand along Row 1
\\ \vspace{\baselineskip} = (a + b + c)[(-b)$ \{ $ c -b$ \} $ -(1)$ \{ $ -c\textsuperscript{2} -(-ab)$ \} $ + a$ \{ $ -c -(-a)$ \} $ ]\\ \\ \vspace{\baselineskip}= (a + b + c)(-bc + b\textsuperscript{2} + c\textsuperscript{2} -ab -ac + a\textsuperscript{2})\\ \\ \vspace{\baselineskip}= a(-bc + b\textsuperscript{2} + c\textsuperscript{2} -ab -ac + a\textsuperscript{2}) + b(-bc + b\textsuperscript{2} + c\textsuperscript{2} -ab -ac + a\textsuperscript{2}) + c(-bc + b\textsuperscript{2} + c\textsuperscript{2} -ab -ac + a\textsuperscript{2})\\


\\ \\ \vspace{\baselineskip}= -abc + ab\textsuperscript{2} + ac\textsuperscript{2} -a\textsuperscript{2}b -a\textsuperscript{2}c + a\textsuperscript{3} -b\textsuperscript{2}c + b\textsuperscript{3} + bc\textsuperscript{2} -ab\textsuperscript{2} -abc + a\textsuperscript{2}b -bc\textsuperscript{2} + b\textsuperscript{2}c + c\textsuperscript{3} -abc -ac\textsuperscript{2}+ a\textsuperscript{2}c\\ \\ \vspace{\baselineskip}= a\textsuperscript{3} + b\textsuperscript{3} + c\textsuperscript{3} -3abc\\

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