#### Need Solution for R.D.Sharma Maths Class 12 Chapter determinants  Exercise 5.2 Question 36 Maths Textbook Solution.

Answer:$(a b+b c+c a)^{3}$

Hint Use determinant formula

Given:$\left|\begin{array}{ccc} -b c & b^{2}+b c & c^{2}+a b \\ a^{2}+a c & -a c & c^{2}+a c \\ a^{2}+a b & b^{2}+a b & -a b \end{array}\right|=(a b+b c+c a)^{3}$

Solution:

\begin{aligned} &\text { L.H.S }\left|\begin{array}{ccc} -b c & b^{2}+b c & c^{2}+a b \\ a^{2}+a c & -a c & c^{2}+a c \\ a^{2}+a b & b^{2}+a b & -a b \end{array}\right| \\ &R_{1} \rightarrow a R_{1}, R_{2} \rightarrow b R_{2}, R_{3} \rightarrow c R_{3} \\ &=\frac{1}{a b c}\left|\begin{array}{ccc} -a b c & a b^{2}+a b c & a c^{2}+a b c \\ a^{2}+a b c & -a b c & c^{2}+a b c \\ a^{2}+a b c & b^{2} c+a b c & -a b c \end{array}\right| \end{aligned}

\begin{aligned} &\text { Common a from } \mathrm{C}_{1}, b \text { from } \mathrm{C}_{2} \text { and } \mathrm{c} \text { from } \mathrm{C}_{3}\\ &=\frac{a b c}{a b c}\left|\begin{array}{ccc} b c & a b+a c & a c+a b \\ a b+b c & -a c & b c+a b \\ a c+b c & b c+a c & -a b \end{array}\right|\\ &\text { Use } R_{1} \rightarrow R_{1}+R_{2}+R_{3}\\ &=\left|\begin{array}{ccc} a b+b c+a c & a b+b c+a c & a b+b c+a c \\ a b+b c & -a c & b c+a b \\ a c+b c & b c+a c & -a b \end{array}\right| \end{aligned}

\begin{aligned} &(a b+b c+a c) \text { common from } \mathrm{R}_{1} \\ &=(a b+b c+a c)\left|\begin{array}{ccc} 1 & 1 & 1 \\ a b+b c & -a c & b c+a b \\ a c+b c & b c+a c & -a b \end{array}\right| \\ &\text { Now } \mathrm{C}_{2} \rightarrow C_{2}-C_{1} \text { and } \mathrm{C}_{3} \rightarrow C_{3}-C_{1} \\ &=(a b+b c+a c) \mid \begin{array}{ccc} 1 & 0 & 0 \\ a b+b c & -(a b+b c+a c) & a b+b c+a c \\ a c+b c & 0 & -(a b+b c+a c) \end{array} \end{aligned}

\begin{aligned} &(a b+b c+a c) \text { common from } \mathrm{C}_{2} \& C_{3}\\ &=(a b+b c+a c)^{3}\left|\begin{array}{ccc} 1 & 0 & 0 \\ a b+b c & -1 & 1 \\ a c+b c & 0 & -1 \end{array}\right|\\ &\text { Expanding w.r.t } \mathrm{R}_{1}\\ &=(a b+b c+a c)^{3}\{1(1-0)+0\}\\ &=(a b+b c+a c)^{3}\\ &=R \cdot H \cdot S \end{aligned}