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Need Solution for R.D.Sharma Maths Class 12 Chapter 5 determinants  Exercise 5.4 Question 21 Maths Textbook Solution.

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Answer: \mathrm{x}=1, \mathrm{y}=\frac{2}{7}, \mathrm{z}=\frac{2}{7} \text { and } \mathrm{w}=\frac{-1}{7}

Hint: Use Cramer’s rule to solve a system of linear equations

Given:

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

Solution:

Solving determinant,

|\mathrm{A}|=\left|\begin{array}{cccc} 2 & 0 & -3 & 1 \\ 1 & -1 & 0 & 2 \\ 0 & -3 & 1 & 1 \\ 1 & 1 & 1 & 0 \end{array}\right|

\begin{aligned} C_{2} & \rightarrow C_{2}-C_{1} \\ C_{3} & \rightarrow C_{3}-C_{1} \end{aligned}

=\left|\begin{array}{cccc} 2 & -2 & -5 & 1 \\ 1 & -2 & -1 & 2 \\ 0 & -3 & 1 & 1 \\ 1 & 0 & 0 & 0 \end{array}\right|

Expanding along R_{1},

=-1\left|\begin{array}{ccc} -2 & -5 & 1 \\ -2 & -1 & 2 \\ -3 & 1 & 1 \end{array}\right|

           \begin{aligned} &C_{2} \rightarrow C_{2}-C_{3} \\ &C_{1} \rightarrow C_{1}+3 C_{3} \end{aligned}

    \begin{aligned} &=-1\left|\begin{array}{ccc} 1 & -6 & 1 \\ 4 & -3 & 2 \\ 0 & 0 & 1 \end{array}\right| \\ &=-1(-3+24) \\ &=-21 \end{aligned}

\begin{aligned} &\mathrm{D}_{\mathrm{x}}=\left|\begin{array}{cccc} 1 & 0 & -3 & 1 \\ 1 & -1 & 0 & 2 \\ 1 & -3 & 1 & 1 \\ 1 & 1 & 1 & 0 \end{array}\right|=-21 \\ &\mathrm{D}_{\mathrm{y}}=\left|\begin{array}{cccc} 2 & 1 & 1 & 0 \end{array}\right|=-6 \\ &\mathrm{D}_{z}=\left|\begin{array}{cccc} 2 & 1 & -3 & 1 \\ 1 & 1 & 0 & 2 \\ 0 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \\ 1 & -1 & 1 & 2 \\ 0 & -3 & 1 & 1 \\ 1 & 1 & 1 & 0 \end{array}\right|=-6 \\ &\mathrm{D}_{\mathrm{w}}=\left|\begin{array}{cccc} 2 & 0 & -3 & 1 \\ 1 & -1 & 0 & 1 \\ 0 & -3 & 1 & 1 \\ 1 & 1 & 1 & 1 \end{array}\right|=3 \end{aligned}

By Cramer’s rule,

\begin{aligned} &\Rightarrow x=\frac{D_{x}}{D}=\frac{-21}{-21}=1 \\ &\Rightarrow y=\frac{D_{y}}{D}=\frac{-6}{-21}=\frac{2}{7} \\ &\Rightarrow z=\frac{D_{z}}{D}=\frac{-6}{-21}=\frac{2}{7} \\ &\Rightarrow w=\frac{D_{w}}{D}=\frac{3}{-21}=\frac{-1}{7} \end{aligned}

Concept: Solving matrix of order 4x4 (Elementary row and column operations)

 

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