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)}](https://www.tinkutara.com/question/Q129964.png)
$$\mathrm{A}\:=\:\begin{bmatrix}{\mathrm{1}}&{\mathrm{1}}&{−\mathrm{15}}&{\mathrm{4}}\\{\mathrm{16}}&{−\mathrm{2}}&{−\mathrm{3}}&{\mathrm{1}}\\{\mathrm{1}}&{−\mathrm{1}}&{\mathrm{3}}&{\mathrm{17}}\\{\mathrm{2}}&{−\mathrm{14}}&{\mathrm{3}}&{\mathrm{2}}\end{bmatrix} \\ $$$$\mathrm{det}\left(\mathrm{A}\right)\:=\:−\mathrm{56460}\:\neq\:\mathrm{0} \\ $$$$\mathrm{A}_{\mathrm{1}} \:=\:\begin{bmatrix}{−\mathrm{2}}&{\mathrm{1}}&{−\mathrm{15}}&{\mathrm{4}}\\{\mathrm{2}}&{−\mathrm{2}}&{−\mathrm{3}}&{\mathrm{1}}\\{\mathrm{9}}&{−\mathrm{1}}&{\mathrm{3}}&{\mathrm{17}}\\{−\mathrm{8}}&{−\mathrm{14}}&{\mathrm{3}}&{\mathrm{2}}\end{bmatrix} \\ $$$$\mathrm{det}\left(\mathrm{A}_{\mathrm{1}} \right)\:=\:−\mathrm{14115} \\ $$$$\mathrm{X}_{\mathrm{1}} \:=\:\frac{\mathrm{det}\left(\mathrm{A}_{\mathrm{1}} \right)}{\mathrm{det}\left(\mathrm{A}\right)}\:=\:\frac{\mathrm{14115}}{\mathrm{56460}}\:=\:\frac{\mathrm{1}}{\mathrm{4}} \\ $$$$\mathrm{A}_{\mathrm{2}} \:=\:\begin{bmatrix}{\mathrm{1}}&{−\mathrm{2}}&{−\mathrm{15}}&{\mathrm{4}}\\{\mathrm{16}}&{\mathrm{2}}&{−\mathrm{3}}&{\mathrm{1}}\\{\mathrm{1}}&{\mathrm{9}}&{\mathrm{3}}&{\mathrm{17}}\\{\mathrm{2}}&{−\mathrm{8}}&{\mathrm{3}}&{\mathrm{2}}\end{bmatrix} \\ $$$$\mathrm{det}\left(\mathrm{A}_{\mathrm{2}} \right)\:=\:−\mathrm{42345} \\ $$$$\mathrm{X}_{\mathrm{2}} \:=\:\frac{\mathrm{det}\left(\mathrm{A}_{\mathrm{2}} \right)}{\mathrm{det}\left(\mathrm{A}\right)}\:=\:\frac{\mathrm{42345}}{\mathrm{56460}}\:=\:\frac{\mathrm{3}}{\mathrm{4}} \\ $$$$\mathrm{A}_{\mathrm{3}} \:=\:\begin{bmatrix}{\mathrm{1}}&{\mathrm{1}}&{−\mathrm{2}}&{\mathrm{4}}\\{\mathrm{16}}&{−\mathrm{2}}&{\mathrm{2}}&{\mathrm{1}}\\{\mathrm{1}}&{−\mathrm{1}}&{\mathrm{9}}&{\mathrm{17}}\\{\mathrm{2}}&{−\mathrm{14}}&{−\mathrm{8}}&{\mathrm{2}}\end{bmatrix} \\ $$$$\mathrm{det}\left(\mathrm{A}_{\mathrm{3}} \right)\:=\:−\mathrm{18820} \\ $$$$\mathrm{X}_{\mathrm{3}} \:=\:\frac{\mathrm{det}\left(\mathrm{A}_{\mathrm{3}} \right)}{\mathrm{det}\left(\mathrm{A}\right)}\:=\:\frac{\mathrm{18820}}{\mathrm{56460}}\:=\:\frac{\mathrm{1}}{\mathrm{3}} \\ $$$$\mathrm{A}_{\mathrm{4}} \:=\:\begin{bmatrix}{\mathrm{1}}&{\mathrm{1}}&{−\mathrm{15}}&{−\mathrm{2}}\\{\mathrm{16}}&{−\mathrm{2}}&{−\mathrm{3}}&{\mathrm{2}}\\{\mathrm{1}}&{−\mathrm{1}}&{\mathrm{3}}&{\mathrm{9}}\\{\mathrm{2}}&{−\mathrm{14}}&{\mathrm{3}}&{−\mathrm{8}}\end{bmatrix} \\ $$$$\mathrm{det}\left(\mathrm{A}_{\mathrm{4}} \right)\:=\:−\mathrm{28230} \\ $$$$\mathrm{X}_{\mathrm{4}} \:=\:\frac{\mathrm{det}\left(\mathrm{A}_{\mathrm{4}} \right)}{\mathrm{det}\left(\mathrm{A}\right)}\:=\:\frac{\mathrm{28230}}{\mathrm{56460}}\:=\:\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\left\{\mathrm{X}_{\mathrm{1}} ,\:\mathrm{X}_{\mathrm{2}} ,\:\mathrm{X}_{\mathrm{3}} ,\:\mathrm{X}_{\mathrm{4}} \right\}\:=\:\left\{\frac{\mathrm{1}}{\mathrm{4}},\:\frac{\mathrm{3}}{\mathrm{4}},\:\frac{\mathrm{1}}{\mathrm{3}},\:\frac{\mathrm{1}}{\mathrm{2}}\right\} \\ $$