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  • Prove that is divisible by a + b + c and find the quotient.

Prove that \left|\begin{array}{ccc} \mathrm{bc}-\mathrm{a}^{2} & \mathrm{ca}-\mathrm{b}^{2} & \mathrm{ab}-\mathrm{c}^{2} \\ \mathrm{ca}-\mathrm{b}^{2} & \mathrm{ab}-\mathrm{c}^{2} & \mathrm{bc}-\mathrm{a}^{2} \\ \mathrm{ab}-\mathrm{c}^{2} & \mathrm{bc}-\mathrm{a}^{2} & \mathrm{ca}-\mathrm{b}^{2} \end{array}\right| is divisible by a + b + c and find the quotient.

Answers (1)

\left|\begin{array}{ccc} \mathrm{bc}-\mathrm{a}^{2} & \mathrm{ca}-\mathrm{b}^{2} & \mathrm{ab}-\mathrm{c}^{2} \\ \mathrm{ca}-\mathrm{b}^{2} & \mathrm{ab}-\mathrm{c}^{2} & \mathrm{bc}-\mathrm{a}^{2} \\ \mathrm{ab}-\mathrm{c}^{2} & \mathrm{bc}-\mathrm{a}^{2} & \mathrm{ca}-\mathrm{b}^{2} \end{array}\right| is given.

Apply,R_1 \rightarrow R_1 - R_2,

\\\begin{aligned} &=\left|\begin{array}{ccc} \mathrm{b} c-\mathrm{a}^{2}-\mathrm{ca}+\mathrm{b}^{2} & \mathrm{ca}-\mathrm{b}^{2}-\mathrm{ab}+\mathrm{c}^{2} & \mathrm{ab}-\mathrm{c}^{2}-\mathrm{bc}+\mathrm{a}^{2} \\ \mathrm{ca}-\mathrm{b}^{2} & \mathrm{ab}-\mathrm{c}^{2} & \mathrm{bc}-\mathrm{a}^{2} \\ \mathrm{ab}-\mathrm{c}^{2} & \mathrm{bc}-\mathrm{a}^{2} & \mathrm{ca}-\mathrm{b}^{2} \end{array}\right|\\ &=\left|\begin{array}{ccc} (b c-c a)+\left(b^{2}-a^{2}\right) & (c a-a b)+\left(c^{2}-b^{2}\right) & (a b-b c)+\left(a^{2}-c^{2}\right) \\ c a-b^{2} & a b-c^{2} & b c-a^{2} \\ a b-c^{2} & b c-a^{2} & c a-b^{2} \end{array}\right|\\ &=\left|\begin{array}{ccc} c(b-a)+(b-a)(b+a) & a(c-b)+(c-b)(c+b) & b(a-c)+(a-c)(a+c) \\ c a-b^{2} & a b-c^{2} & b c-a^{2} \\ a b-c^{2} & b c-a^{2} & c a-b^{2} \end{array}\right|\\ &=\left|\begin{array}{ccc} (b-a)(c+b+a) & (c-b)(a+c+b) & (a-c)(b+a+c) \\ c a-b^{2} & a b-c^{2} & b c-a^{2} \\ a b-c^{2} & b c-a^{2} & c a-b^{2} \end{array}\right| \end{aligned}

Take (a+b+c) common from Column 1

\\\begin{aligned} &=(a+b+c)\left|\begin{array}{ccc} (b-a) & (c-b) & (a-c) \\ c a-b^{2} & a b-c^{2} & b c-a^{2} \\ a b-c^{2} & b c-a^{2} & c a-b^{2} \end{array}\right|\\ &\text { Apply } \mathrm{R}_{2} \rightarrow \mathrm{R}_{2}-\mathrm{R}_{3}\\ &=(a+b+c)\left|\begin{array}{ccc} (b-a) & (c-b) & (a-c) \\ c a-b^{2}-a b+c^{2} & a b-c^{2}-b c+a^{2} & b c-a^{2}-c a+b^{2} \\ a b-c^{2} & b c-a^{2} & c a-b^{2} \end{array}\right|\\ &=(a+b+c)\left|\begin{array}{ccc} (b-a) & (c-b) & (a-c) \\ (c-b)(a+b+c) & (a-c)(a+b+c) & (b-a)(a+b+c) \\ a b-c^{2} & b c-a^{2} & c a-b^{2} \end{array}\right| \end{aligned}

Take (a+b+c) common from Column 2

\\\begin{aligned} &=(a+b+c)^{2}\left|\begin{array}{ccc} b-a+c-b+a-c & (c-b) & (a-c) \\ c-b+a-c+b-a & (a-c) & (b-a) \\ a b-c^{2}+b c-a^{2}+c a-b^{2} & b c-a^{2} & c a-b^{2} \end{array}\right|\\ &=(a+b+c)^{2}\left|\begin{array}{ccc} 0 & (c-b) & (a-c) \\ 0 & (a-c) & (b-a) \\ a b+b c+c a-\left(a^{2}+b^{2}+c^{2}\right) & b c-a^{2} & c a-b^{2} \end{array}\right| \end{aligned}

Expand along Column 1

 

\\= (a + b + c)\textsuperscript{2}[ \{ab + bc + ca -(a\textsuperscript{2} + b\textsuperscript{2} + c\textsuperscript{2})$ \} \{ $ (c -b)(b -a) -(a -c)\textsuperscript{2} \} ]\\ \\ \vspace{\baselineskip}= (a + b + c)\textsuperscript{2}[\{ab + bc + ca -(a\textsuperscript{2} + b\textsuperscript{2} + c\textsuperscript{2})$\} \{ $ (cb -ac -b\textsuperscript{2} + ab -(a + c\textsuperscript{2} -2ac)$ \} ]\\ \\ \vspace{\baselineskip}= (a + b + c)\textsuperscript{2}[\{ab + bc + ca -(a\textsuperscript{2} + b\textsuperscript{2} + c\textsuperscript{2})$\} \{ $ (cb -ac -b\textsuperscript{2} + ab -a - c\textsuperscript{2} + 2ac) \} ]\\ \\ \vspace{\baselineskip}= (a + b + c)\textsuperscript{2}[\{ab + bc + ca -(a\textsuperscript{2} + b\textsuperscript{2} + c\textsuperscript{2})$\} \{ $ ac + bc + ab -(a\textsuperscript{2} + b\textsuperscript{2} + c\textsuperscript{2})$ \} ]\\ \\

 

\\\vspace{\baselineskip}=(a + b + c)\textsuperscript{2}[ab + bc + ca -(a\textsuperscript{2} + b\textsuperscript{2} + c\textsuperscript{2})]\textsuperscript{2}\\

\\\vspace{\baselineskip}=(a + b + c)(a + b + c)[ab + bc + ca -(a\textsuperscript{2} + b\textsuperscript{2} + c\textsuperscript{2})]^2\\

The determinant is divisible by (a+b+c) and the quotient is \\(a + b + c)[ab + bc + ca -(a\textsuperscript{2} + b\textsuperscript{2} + c\textsuperscript{2})]^2\\

 

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