Matrices and Determinants 4 Question 12

12. If the system of linear equations

$ x-4 y+7 z=g, 3 y-5 z=h,-2 x+5 y-9 z=k $

is consistent, then

(2019 Main, 9 Jan II)

(a) $2 g+h+k=0$

(b) $g+2 h+k=0$

(c) $g+h+k=0$

(d) $g+h+2 k=0$

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Answer:

Correct Answer: 12. (a)

Solution:

  1. (a) Here, $D=\left|\begin{array}{ccc}1 & -4 & 7 \\ 0 & 3 & -5 \\ -2 & 5 & -9\end{array}\right|$

$ =1(-27+25)+4(0-10)+7(0+6) $

[expanding along $R_{1}$ ]

$ =-2-40+42=0 $

$\therefore$ The system of linear equations have infinite many solutions.

$[\because$ system is consistent and does not have unique solution as $D=0]$

$\Rightarrow \quad D_{1}=D_{2}=D_{3}=0$

Now, $D_{1}=0 \Rightarrow\left|\begin{array}{ccc}g & -4 & 7 \\ h & 3 & -5 \\ k & 5 & -9\end{array}\right|=0$

$\Rightarrow g(-27+25)+4(-9 h+5 k)+7(5 h-3 k)=0$

$\Rightarrow-2 g-36 h+20 k+35 h-21 k=0$

$\Rightarrow-2 g-h-k=0 \Rightarrow 2 g+h+k=0$



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