Need Solution for R.D.Sharma Maths Class 12 Chapter 7 Solution of Simultaneous Linear Equation Exercise Fill in the blank Question 5 Maths Textbook Solution.

Answer $\rightarrow k=\pm 1$

Given $\rightarrow$ The system of equations $x-k y-z=0, k x-y-z=0 \text { and } x+y-z=0$ has non-zero solution

To find $\rightarrow$ We have to find out the value of $k$.

Hint $\rightarrow$ The system has a non-zero solution if $\left | A \right |=0$

Solution $\rightarrow$ Given system of linear equation

\begin{aligned} &x-k y-z=0 \\ &k x-y-z=0 \\ &x+y-z=0 \end{aligned}

For the given system of equations we have,

$\Rightarrow A=\left|\begin{array}{ccc} 1 & -k & -1 \\ k & -1 & -1 \\ 1 & 1 & -1 \end{array}\right|$

We know, if system of equation has a non-zero solution then $\left | A \right |=0$

$\text { Now, }|A|=\left|\begin{array}{ccc} 1 & -k & -1 \\ k & -1 & -1 \\ 1 & 1 & -1 \end{array}\right|=0$

\begin{aligned} &\Rightarrow 1(1+1)+k(-k+1)-1(k+1)=0 \\ &\Rightarrow 2-k^{2}+k-k-1=0 \\ &\Rightarrow k^{2}=1 \\ &\Rightarrow k=\pm 1 \end{aligned}