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Provide solution for RD Sharma maths class 12 chapter Solution of Simultaneous Linear Equation exercise 7.1 question 2 subquestion (viii)

Answers (1)

Answer:

\begin{aligned} & x=1 \quad, \quad \mathrm{y}=1 \quad, \quad z=-1 \end{aligned}

Given:

\begin{aligned} &2 x+y+z=2 \\ &x+3 y-z=5 \\ &3 x+y-2 z=6 \end{aligned}

Hint:

X=A-1B is used to solve this problem. First we find the determinant of matrix A i.e |A| then will find the co-factor of matrix A, take it’s transpose, and that will be Adj A using Adj A calculate A-1.

Solution:

        \begin{aligned} &{\left[\begin{array}{ccc} 2 & 1 & 1 \\ 1 & 3 & -1 \\ 3 & 1 & -2 \end{array}\right]\left[\begin{array}{l} x \\ y \\ z \end{array}\right]=\left[\begin{array}{l} 2 \\ 5 \\ 6 \end{array}\right]} \\ &A X=B \end{aligned}

        \begin{aligned} |A|=\left|\begin{array}{ccc} 2 & 1 & 1 \\ 1 & 3 & -1 \\ 3 & 1 & -2 \end{array}\right|=& 2(-6+1)-1(-2+3)+1(1-9) \\ &=10-1-8 \\ &=-19 \end{aligned}

        C_{i j}\; be\; the\; co-\! f\! actor\; o\! f\; the\; elements\; a_{i j}\; in \; A=\left[a_{i j}\right].

        \begin{aligned} &C_{11}=(-1)^{1+1}\left|\begin{array}{ll} 3 & -1 \\ 1 & -2 \end{array}\right|=-5 \quad, \quad C_{12}=(-1)^{1+2}\left|\begin{array}{cc} 1 & -1 \\ 3 & -2 \end{array}\right|=-1 \\ &C_{13}=(-1)^{1+3}\left|\begin{array}{ll} 1 & 3 \\ 3 & 1 \end{array}\right|=-8 \quad, \quad C_{21}=(-1)^{2+1}\left|\begin{array}{cc} 1 & 1 \\ 1 & -2 \end{array}\right|=3 \end{aligned}

        \begin{aligned} &C_{22}=(-1)^{2+2}\left|\begin{array}{cc} 2 & -1 \\ 3 & -2 \end{array}\right|=-7 \quad, \quad C_{23}=(-1)^{2+3}\left|\begin{array}{cc} 2 & 1 \\ 3 & 1 \end{array}\right|=1 \\ &C_{31}=(-1)^{3+1}\left|\begin{array}{cc} 1 & 1 \\ 3 & -1 \end{array}\right|=-4 \quad, \quad C_{32}=(-1)^{3+2}\left|\begin{array}{cc} 2 & 1 \\ 1 & -1 \end{array}\right|=3 \\ &C_{33}=(-1)^{3+3}\left|\begin{array}{ll} 2 & 1 \\ 1 & 3 \end{array}\right|=5 \end{aligned}

        \begin{aligned} \operatorname{adj} A &=\left[\begin{array}{ccc} -5 & -1 & -8 \\ 3 & -7 & 1 \\ -4 & 3 & 5 \end{array}\right]^{T} \\ &=\left[\begin{array}{ccc} -5 & 3 & -4 \\ -1 & -7 & 3 \\ -8 & 1 & 5 \end{array}\right] \end{aligned}

        \begin{aligned} A^{-1} &=\frac{1}{|A|} \operatorname{adjA} \\ &=\frac{1}{-19}\left[\begin{array}{ccc} -5 & 3 & -4 \\ -1 & -7 & 3 \\ -8 & 1 & 5 \end{array}\right] \end{aligned}

        \begin{aligned} X &=A^{-1} B \\ &=\frac{1}{-19}\left[\begin{array}{ccc} -5 & 3 & -4 \\ -1 & -7 & 3 \\ -8 & 1 & 5 \end{array}\right]\left[\begin{array}{l} 2 \\ 5 \\ 6 \end{array}\right] \\ &=\frac{1}{-19}\left[\begin{array}{c} -10+15-24 \\ -2-35+18 \\ -16+5+30 \end{array}\right] \end{aligned}

        \begin{aligned} &=\frac{1}{-19}\left[\begin{array}{c} -19 \\ -19 \\ 19 \end{array}\right] \\ &{\left[\begin{array}{l} x \\ y \\ z \end{array}\right]=\left[\begin{array}{c} 1 \\ 1 \\ -1 \end{array}\right]} \\ &x=1 \quad, \quad y=1 \quad, \quad z=-1 \end{aligned}

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Gurleen Kaur

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