Provide Solution For  R.D. Sharma Maths Class 12 Chapter determinants Exercise 5.2 Question 2 Sub Question 5 Maths Textbook Solution.

Answer:$\left|\begin{array}{lll} a+b & 2 a+b & 3 a+b \\ 2 a+b & 3 a+b & 4 a+b \\ 4 a+b & 5 a+b & 6 a+b \end{array}\right|=0$

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

Given:$\left|\begin{array}{lll} a+b & 2 a+b & 3 a+b \\ 2 a+b & 3 a+b & 4 a+b \\ 4 a+b & 5 a+b & 6 a+b \end{array}\right|$

Solution:$\left|\begin{array}{lll} a+b & 2 a+b & 3 a+b \\ 2 a+b & 3 a+b & 4 a+b \\ 4 a+b & 5 a+b & 6 a+b \end{array}\right|$

\begin{aligned} &\text { Applying } \mathrm{C}_{2} \rightarrow C_{2}-C_{1} \text { and } \mathrm{C}_{3} \rightarrow C_{3}-C_{2} \\ &=\left|\begin{array}{lll} a+b & (2 a+b)-(a+b) & (3 a+b)-(2 a+b) \\ 2 a+b & (3 a+b)-(2 a+b) & (4 a+b)-(3 a+b) \\ 4 a+b & (5 a+b)-(4 a+b) & (6 a+b)-(5 a+b) \end{array}\right| \\ &=\left|\begin{array}{lll} a+b & 2 a+b-a-b & 3 a+b-2 a-b \\ 2 a+b & 3 a+b-2 a-b & 4 a+b-3 a-b \\ 4 a+b & 5 a+b-4 a-b & 6 a+b-5 a-b \end{array}\right| \\ &=\left|\begin{array}{lll} a+b & a & a \\ 2 a+b & a & a \\ 4 a+b & a & a \end{array}\right| \end{aligned}

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

The value of the determinant is zero

$=0 \quad\left(\because C_{2}=C_{3}\right)$

Hence$\left|\begin{array}{lll} a+b & 2 a+b & 3 a+b \\ 2 a+b & 3 a+b & 4 a+b \\ 4 a+b & 5 a+b & 6 a+b \end{array}\right|=0$