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Using properties of determinants prove that: $\left|\begin{array}{lll} b+c & a-b & a \\ c+a & b-c & b \\ a+b & c-a & c \end{array}\right|=3 a b c-a^{3}-b^{3}-c^{3}$
Using properties of determinants prove that: $\left|\begin{array}{lll} b+c & a-b & a \\ c+a & b-c & b \\ a+b & c-a & c \end{array}\right|=3 a b c-a^{3}-b^{3}-c^{3}$
CBSE | Class 12 | Determinants | Exercise 6B | Maths | RS Aggarwal
Using properties of determinants prove that: $\left|\begin{array}{lll} b+c & a-b & a \\ c+a & b-c & b \\ a+b & c-a & c \end{array}\right|=3 a b c-a^{3}-b^{3}-c^{3}$

Solution:

$\begin{array}{l}
\left|\begin{array}{lll}
\mathrm{b}+\mathrm{c} & \mathrm{a}-\mathrm{b} & \mathrm{a} \\
\mathrm{c}+\mathrm{a} & \mathrm{b}-\mathrm{c} & \mathrm{b} \\
\mathrm{a}+\mathrm{b} & \mathrm{c}-\mathrm{a} & \mathrm{c}
\end{array}\right| \\
=\left|\begin{array}{ccc}
2(\mathrm{a}+\mathrm{b}+\mathrm{c}) & 0 & \mathrm{a}+\mathrm{b}+\mathrm{c} \\
\mathrm{c}+\mathrm{a} & \mathrm{b}-\mathrm{c} & \mathrm{b} \\
\mathrm{a}+\mathrm{b} & \mathrm{c}-\mathrm{a} & \mathrm{c}
\end{array}\right|\left[\mathrm{R}_{1}^{\prime}=\mathrm{R}_{1}+\mathrm{R}_{2}+\mathrm{R}_{3}\right] \\
=(\mathrm{a}+\mathrm{b}+\mathrm{c})\left|\begin{array}{ccc}
2 & 0 & 1 \\
\mathrm{c}+\mathrm{a}+\mathrm{b} & \mathrm{b}-\mathrm{c} & \mathrm{b} \\
\mathrm{c}-\mathrm{a} & \mathrm{c}
\end{array}\right|\left[\mathrm{R}_{1}^{\prime}=\mathrm{R}_{1} /(\mathrm{a}+\mathrm{b}+\mathrm{c})\right] \\
=(\mathrm{a}+\mathrm{b}+\mathrm{c})[2(\mathrm{~b}-\mathrm{c}) \mathrm{c}-\mathrm{b}(\mathrm{c}-\mathrm{a})+(\mathrm{c}+\mathrm{a})(\mathrm{c}-\mathrm{a})-(\mathrm{a}+\mathrm{b})(\mathrm{b}-\mathrm{c})][\text { expansion by first row }] \\
=(\mathrm{a}+\mathrm{b}+\mathrm{c})\left(2 \mathrm{bc}-2 \mathrm{c}^{2}-\mathrm{bc}+\mathrm{ab}+\mathrm{c}^{2}-\mathrm{a}^{2}-\mathrm{ab}-\mathrm{b}^{2}+\mathrm{ac}+\mathrm{bc}\right. \\
=(\mathrm{a}+\mathrm{b}+\mathrm{c})\left(\mathrm{ab}+\mathrm{bc}+\mathrm{ac}-\mathrm{a}^{2}-\mathrm{b}^{2}-\mathrm{c}^{2}\right) \\
=3 \mathrm{abc}-\mathrm{a}^{3}-\mathrm{b}^{3}-\mathrm{c}^{3}
\end{array}$

i

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