#### Provide solution for RD Sharma maths Class 12 Chapter 5 Determinants Exercise VSQ Question 51 maths textbook solution.

Hint: Here we use basic concept of determinant of matrix

Given: $\left|\begin{array}{ccc} x+y & y+z & z+x \\ z & x & y \\ -3 & -3 & -3 \end{array}\right|$

Solution :

$\rightarrow$ Let's perform row operations

\begin{aligned} &\mathrm{R}_{1} \rightarrow \mathrm{R}_{1}+\mathrm{R}_{2} \\ &=\left|\begin{array}{ccc} x+y+z & x+y+z & x+y+z \\ z & x & y \\ -3 & -3 & -3 \end{array}\right| \end{aligned}

$\rightarrow$ Let's take $(x+y+z)$ common from $R_{1}$ and $-3$ from $R_{3}$

$=(x+y+z)(-3)\left|\begin{array}{lll} 1 & 1 & 1 \\ z & x & y \\ 1 & 1 & 1 \end{array}\right|$

$\rightarrow$ Here we exan clearly see that two rows are identical.

So, it's determinate is 0.