#### Provide Solution For  R.D.Sharma Maths Class 12 Chapter 5 determinants Exercise 5.4 Question 24 Maths Textbook Solution.

Answer: $\mathrm{x}=\infty, \mathrm{y}=\infty \text { and } \mathrm{z}=\infty$

Hint: Solving determinant gives zero.

Given:

\begin{aligned} &3 x-y+2 z=3 \\ &2 x+y+3 z=5 \\ &x-2 y-z=1 \end{aligned}

Solution:

By Cramer’s rule:

Solving determinant,

$|\mathrm{A}|=\left|\begin{array}{ccc} 3 & -1 & 2 \\ 2 & 1 & 3 \\ 1 & -2 & -1 \end{array}\right|$

Expanding along $1^{st}$ row,

\begin{aligned} &=3(-1+6)+1(-2-3)+2(-4-1) \\ &=3(5)+1(-5)+2(-5) \\ &=15-5-10 \\ &=0 \end{aligned}

\begin{aligned} \mathrm{D} &=0 \\ \mathrm{D}_{x} &=\left|\begin{array}{ccc} 3 & -1 & 2 \\ 5 & 1 & 3 \\ 1 & -2 & -1 \end{array}\right| \\ &=3(-1+6)+1(-5-3)+2(-10-1) \\ &=3(5)+1(-8)+2(-11) \\ &=15-8-22 \\ &=-15 \neq 0 \end{aligned}

By Cramer’s rule,

\begin{aligned} &\Rightarrow x=\frac{D_{x}}{D}=\frac{-19}{0}=\infty\\ &\text { Since, } \mathrm{D}=0 \text { and } \mathrm{D}_{x}, \mathrm{D}_{\mathrm{y}} \text { and } \mathrm{D}_{z} \neq 0 \end{aligned}

$\therefore$ Linear equations are inconsistent.

Concept: Solving matrix of order 3x3 by solving linear equations

Note: When D = 0, there is either no solution or infinite solutions.