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Question Number 15233 by tawa tawa last updated on 08/Jun/17

Find the general solution of the following homogeneous system of  equation    x_1  + x_2  − 2x_3  = 0  3x_1  − x_2  − 6x_3  = 0  −2x_1  + 3x_(2 ) + 4x_3  = 0

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{general}\:\mathrm{solution}\:\mathrm{of}\:\mathrm{the}\:\mathrm{following}\:\mathrm{homogeneous}\:\mathrm{system}\:\mathrm{of} \\ $$$$\mathrm{equation}\:\: \\ $$$$\mathrm{x}_{\mathrm{1}} \:+\:\mathrm{x}_{\mathrm{2}} \:−\:\mathrm{2x}_{\mathrm{3}} \:=\:\mathrm{0} \\ $$$$\mathrm{3x}_{\mathrm{1}} \:−\:\mathrm{x}_{\mathrm{2}} \:−\:\mathrm{6x}_{\mathrm{3}} \:=\:\mathrm{0} \\ $$$$−\mathrm{2x}_{\mathrm{1}} \:+\:\mathrm{3x}_{\mathrm{2}\:} +\:\mathrm{4x}_{\mathrm{3}} \:=\:\mathrm{0} \\ $$

Answered by arnabpapu550@gmail.com last updated on 08/Jun/17

The given equations can be written as the  following matrix,  AX=O  where A=  [(1,1,(−2)),(3,(−1),(−6)),((−2),3,4) ]but det(A)=2≠0  Hence the equations are consistant.  X= ((x_1 ),(x_2 ),(x_3 ) )   and O= ((0),(0),(0) )  ∴ X=A^(−1) .O= ((0),(0),(0) )   ∴x_1 =0, x_2 =0 and x_3 =0  Is it right Tawa Tawa?

$$\mathrm{The}\:\mathrm{given}\:\mathrm{equations}\:\mathrm{can}\:\mathrm{be}\:\mathrm{written}\:\mathrm{as}\:\mathrm{the} \\ $$$$\mathrm{following}\:\mathrm{matrix},\:\:\mathrm{AX}=\mathrm{O} \\ $$$$\mathrm{where}\:\mathrm{A}=\:\begin{bmatrix}{\mathrm{1}}&{\mathrm{1}}&{−\mathrm{2}}\\{\mathrm{3}}&{−\mathrm{1}}&{−\mathrm{6}}\\{−\mathrm{2}}&{\mathrm{3}}&{\mathrm{4}}\end{bmatrix}\mathrm{but}\:\mathrm{det}\left(\mathrm{A}\right)=\mathrm{2}\neq\mathrm{0} \\ $$$$\mathrm{Hence}\:\mathrm{the}\:\mathrm{equations}\:\mathrm{are}\:\mathrm{consistant}. \\ $$$$\mathrm{X}=\begin{pmatrix}{\mathrm{x}_{\mathrm{1}} }\\{\mathrm{x}_{\mathrm{2}} }\\{\mathrm{x}_{\mathrm{3}} }\end{pmatrix}\:\:\:\mathrm{and}\:\mathrm{O}=\begin{pmatrix}{\mathrm{0}}\\{\mathrm{0}}\\{\mathrm{0}}\end{pmatrix} \\ $$$$\therefore\:\mathrm{X}=\mathrm{A}^{−\mathrm{1}} .\mathrm{O}=\begin{pmatrix}{\mathrm{0}}\\{\mathrm{0}}\\{\mathrm{0}}\end{pmatrix}\: \\ $$$$\therefore\mathrm{x}_{\mathrm{1}} =\mathrm{0},\:\mathrm{x}_{\mathrm{2}} =\mathrm{0}\:\mathrm{and}\:\mathrm{x}_{\mathrm{3}} =\mathrm{0} \\ $$$$\mathrm{Is}\:\mathrm{it}\:\mathrm{right}\:\mathrm{Tawa}\:\mathrm{Tawa}? \\ $$

Commented by tawa tawa last updated on 08/Jun/17

I don′t know too, God bless you sir.

$$\mathrm{I}\:\mathrm{don}'\mathrm{t}\:\mathrm{know}\:\mathrm{too},\:\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir}. \\ $$

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