Need Solution for R.D.Sharma Maths Class 12 Chapter 7 Solution of Simultaneous Linear Equation Exercise Fill in the blank Question 1  Maths Textbook Solution.

Answer $\rightarrow a=-1$

Given $\rightarrow$ Here given that $x+a y=0, a z+y=0, a x+z-0$ has infinitely  many solutions.

To find $\rightarrow$ The value of $a$.

Hint $\rightarrow$ Given system of the equation can be written as $AX=0$ then solve to find $a$.

Solution$\rightarrow$  We know that $x+a y=0, a z+y=0, a x+z-0$

Given system of linear equation can be written as $AX=0$ where,

$A=\left[\begin{array}{lll} 1 & a & 0 \\ 0 & 1 & a \\ a & 0 & 1 \end{array}\right], X=\left[\begin{array}{l} x \\ y \\ z \end{array}\right] \text { and } 0=\left[\begin{array}{l} 0 \\ 0 \\ 0 \end{array}\right]$

We know that for infinitely many solution $D=0$

$\Rightarrow\left|\begin{array}{lll} 1 & a & 0 \\ 0 & 1 & a \\ a & 0 & 1 \end{array}\right|=0$

Expanding along row $1$ we get

\begin{aligned} &\Rightarrow 1(1-0)-a\left(0-a^{2}\right)+0(0-a)=0 \\ &\Rightarrow 1+a^{3}=0 \end{aligned}

\begin{aligned} &\Rightarrow a^{3}+1=0 \\ &\Rightarrow a^{3}=-1 \\ &\Rightarrow a=-1 \end{aligned}

This is required value of $a$