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Question-129955




Question Number 129955 by mohammad17 last updated on 21/Jan/21
Answered by Olaf last updated on 21/Jan/21
A =  [(1,1,(−15),4),((16),(−2),(−3),1),(1,(−1),3,(17)),(2,(−14),3,2) ]  det(A) = −56460 ≠ 0  A_1  =  [((−2),1,(−15),4),(2,(−2),(−3),1),(9,(−1),3,(17)),((−8),(−14),3,2) ]  det(A_1 ) = −14115  X_1  = ((det(A_1 ))/(det(A))) = ((14115)/(56460)) = (1/4)  A_2  =  [(1,(−2),(−15),4),((16),2,(−3),1),(1,9,3,(17)),(2,(−8),3,2) ]  det(A_2 ) = −42345  X_2  = ((det(A_2 ))/(det(A))) = ((42345)/(56460)) = (3/4)  A_3  =  [(1,1,(−2),4),((16),(−2),2,1),(1,(−1),9,(17)),(2,(−14),(−8),2) ]  det(A_3 ) = −18820  X_3  = ((det(A_3 ))/(det(A))) = ((18820)/(56460)) = (1/3)  A_4  =  [(1,1,(−15),(−2)),((16),(−2),(−3),2),(1,(−1),3,9),(2,(−14),3,(−8)) ]  det(A_4 ) = −28230  X_4  = ((det(A_4 ))/(det(A))) = ((28230)/(56460)) = (1/2)  {X_1 , X_2 , X_3 , X_4 } = {(1/4), (3/4), (1/3), (1/2)}
$$\mathrm{A}\:=\:\begin{bmatrix}{\mathrm{1}}&{\mathrm{1}}&{−\mathrm{15}}&{\mathrm{4}}\\{\mathrm{16}}&{−\mathrm{2}}&{−\mathrm{3}}&{\mathrm{1}}\\{\mathrm{1}}&{−\mathrm{1}}&{\mathrm{3}}&{\mathrm{17}}\\{\mathrm{2}}&{−\mathrm{14}}&{\mathrm{3}}&{\mathrm{2}}\end{bmatrix} \\ $$$$\mathrm{det}\left(\mathrm{A}\right)\:=\:−\mathrm{56460}\:\neq\:\mathrm{0} \\ $$$$\mathrm{A}_{\mathrm{1}} \:=\:\begin{bmatrix}{−\mathrm{2}}&{\mathrm{1}}&{−\mathrm{15}}&{\mathrm{4}}\\{\mathrm{2}}&{−\mathrm{2}}&{−\mathrm{3}}&{\mathrm{1}}\\{\mathrm{9}}&{−\mathrm{1}}&{\mathrm{3}}&{\mathrm{17}}\\{−\mathrm{8}}&{−\mathrm{14}}&{\mathrm{3}}&{\mathrm{2}}\end{bmatrix} \\ $$$$\mathrm{det}\left(\mathrm{A}_{\mathrm{1}} \right)\:=\:−\mathrm{14115} \\ $$$$\mathrm{X}_{\mathrm{1}} \:=\:\frac{\mathrm{det}\left(\mathrm{A}_{\mathrm{1}} \right)}{\mathrm{det}\left(\mathrm{A}\right)}\:=\:\frac{\mathrm{14115}}{\mathrm{56460}}\:=\:\frac{\mathrm{1}}{\mathrm{4}} \\ $$$$\mathrm{A}_{\mathrm{2}} \:=\:\begin{bmatrix}{\mathrm{1}}&{−\mathrm{2}}&{−\mathrm{15}}&{\mathrm{4}}\\{\mathrm{16}}&{\mathrm{2}}&{−\mathrm{3}}&{\mathrm{1}}\\{\mathrm{1}}&{\mathrm{9}}&{\mathrm{3}}&{\mathrm{17}}\\{\mathrm{2}}&{−\mathrm{8}}&{\mathrm{3}}&{\mathrm{2}}\end{bmatrix} \\ $$$$\mathrm{det}\left(\mathrm{A}_{\mathrm{2}} \right)\:=\:−\mathrm{42345} \\ $$$$\mathrm{X}_{\mathrm{2}} \:=\:\frac{\mathrm{det}\left(\mathrm{A}_{\mathrm{2}} \right)}{\mathrm{det}\left(\mathrm{A}\right)}\:=\:\frac{\mathrm{42345}}{\mathrm{56460}}\:=\:\frac{\mathrm{3}}{\mathrm{4}} \\ $$$$\mathrm{A}_{\mathrm{3}} \:=\:\begin{bmatrix}{\mathrm{1}}&{\mathrm{1}}&{−\mathrm{2}}&{\mathrm{4}}\\{\mathrm{16}}&{−\mathrm{2}}&{\mathrm{2}}&{\mathrm{1}}\\{\mathrm{1}}&{−\mathrm{1}}&{\mathrm{9}}&{\mathrm{17}}\\{\mathrm{2}}&{−\mathrm{14}}&{−\mathrm{8}}&{\mathrm{2}}\end{bmatrix} \\ $$$$\mathrm{det}\left(\mathrm{A}_{\mathrm{3}} \right)\:=\:−\mathrm{18820} \\ $$$$\mathrm{X}_{\mathrm{3}} \:=\:\frac{\mathrm{det}\left(\mathrm{A}_{\mathrm{3}} \right)}{\mathrm{det}\left(\mathrm{A}\right)}\:=\:\frac{\mathrm{18820}}{\mathrm{56460}}\:=\:\frac{\mathrm{1}}{\mathrm{3}} \\ $$$$\mathrm{A}_{\mathrm{4}} \:=\:\begin{bmatrix}{\mathrm{1}}&{\mathrm{1}}&{−\mathrm{15}}&{−\mathrm{2}}\\{\mathrm{16}}&{−\mathrm{2}}&{−\mathrm{3}}&{\mathrm{2}}\\{\mathrm{1}}&{−\mathrm{1}}&{\mathrm{3}}&{\mathrm{9}}\\{\mathrm{2}}&{−\mathrm{14}}&{\mathrm{3}}&{−\mathrm{8}}\end{bmatrix} \\ $$$$\mathrm{det}\left(\mathrm{A}_{\mathrm{4}} \right)\:=\:−\mathrm{28230} \\ $$$$\mathrm{X}_{\mathrm{4}} \:=\:\frac{\mathrm{det}\left(\mathrm{A}_{\mathrm{4}} \right)}{\mathrm{det}\left(\mathrm{A}\right)}\:=\:\frac{\mathrm{28230}}{\mathrm{56460}}\:=\:\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\left\{\mathrm{X}_{\mathrm{1}} ,\:\mathrm{X}_{\mathrm{2}} ,\:\mathrm{X}_{\mathrm{3}} ,\:\mathrm{X}_{\mathrm{4}} \right\}\:=\:\left\{\frac{\mathrm{1}}{\mathrm{4}},\:\frac{\mathrm{3}}{\mathrm{4}},\:\frac{\mathrm{1}}{\mathrm{3}},\:\frac{\mathrm{1}}{\mathrm{2}}\right\} \\ $$

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