#### Provide Solution For  R.D. Sharma Maths Class 12 Chapter 5.2 determinants Exercise MCQs Question 2 sub question 4 Maths Textbook Solution.

Answer: $\left|\begin{array}{lll} \frac{1}{a} & a^{2} & b c \\ \frac{1}{b} & b^{2} & a c \\ \frac{1}{c} & c^{2} & a b \end{array}\right|=0$

Hint: We will try to do any two column or row equal

Given: $\left|\begin{array}{lll} \frac{1}{a} & a^{2} & b c \\ \frac{1}{b} & b^{2} & a c \\ \frac{1}{c} & c^{2} & a b \end{array}\right|$

Solution: $\left|\begin{array}{lll} \frac{1}{a} & a^{2} & b c \\ \frac{1}{b} & b^{2} & a c \\ \frac{1}{c} & c^{2} & a b \end{array}\right|$

\begin{aligned} &\text { Applying } \mathrm{C}_{3} \rightarrow C_{3} \div(a b c) \\ &=a b c\left|\begin{array}{ccc} \frac{1}{a} & a^{2} & \frac{b c}{a b c} \\ \frac{1}{b} & b^{2} & \frac{a c}{a b c} \\ \frac{1}{c} & c^{2} & \frac{a b}{a b c} \end{array}\right| \\ &=a b c\left|\begin{array}{ccc} \frac{1}{a} & a^{2} & \frac{1}{a} \\ \frac{1}{b} & b^{2} & \frac{1}{b} \\ \frac{1}{c} & c^{2} & \frac{1}{c} \end{array}\right| \end{aligned}

If any two rows or columns of a determinant are identical.

The value of the determinant is zero

\begin{aligned} &=0 \times a b c \quad\left(\because C_{1}=C_{3}\right) \\ &=0 \end{aligned}

Hence $\left|\begin{array}{lll} \frac{1}{a} & a^{2} & b c \\ \frac{1}{b} & b^{2} & a c \\ \frac{1}{c} & c^{2} & a b \end{array}\right|=0$