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  • Please solve RD Sharma class 12 chapter 7 Solution of Simultaneous Linear Equation Exercise Fill in the blank Question 6 Maths Textbook Solution.

Please solve RD Sharma class 12 chapter 7 Solution of Simultaneous Linear Equation Exercise Fill in the blank Question 6 Maths Textbook Solution.

Answers (1)

Answer \rightarrow 

Given  \rightarrow The given system of equations \lambda x+y+z=0,-x+\lambda y+z=0 \text { and }-x-y+\lambda z=0has a non-zero solution.

To find \rightarrow We have to find the value of \lambda.

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

Solution \rightarrow  Here system of equations are

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

Then,
\Rightarrow A=\left|\begin{array}{ccc} \lambda & 1 & 1 \\ -1 & \lambda & 1 \\ -1 & -1 & \lambda \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} \lambda & 1 & 1 \\ -1 & \lambda & 1 \\ -1 & -1 & \lambda \end{array}\right|=0

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

In this case \lambda is an imaginary number which is non-existent.

              \Rightarrow \lambda=0

Hence, \lambda=0 is required answer.

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