Solve the system of equations, using matrix method2x+y+z=1, x−2y−z=23, 3y−5z=9

Question

Solve the system of equations, using matrix method2x+y+z=1, x−2y−z=23​, 3y−5z=9

Filo · Accepted Answer

Given system of equations2x+y+z=1x−2y−z=23​3y−5z=9This can be written as AX=Bwhere A=⎣⎡​210​1−23​1−1−5​⎦⎤​,X=⎣⎡​xyz​⎦⎤​,B=⎣⎡​123​9​⎦⎤​Here, ∣A∣=2(10+3)−1(−5−0)+1(3−0)⇒∣A∣=26+5+3=34Since, ∣A∣= 0Hence, the system of equations is consistent and has a unique solution given by X==A−1BA−1=∣A∣adjA​ and adjA=CTC11​=(−1)1+1 ∣∣​−23​−1−5​∣∣​⇒C11​=10+3=13C12​=(−1)1+2 ∣∣​10​−1−5​∣∣​⇒C12​=−(−5−0)=5C13​=(−1)1+3 ∣∣​10​−23​∣∣​⇒C13​=3−0=3C21​=(−1)2+1 ∣∣​13​1−5​∣∣​⇒C21​=−(−5−3)=8C22​=(−1)2+2 ∣∣​20​1−5​∣∣​⇒C22​=−10−0=−10C23​=(−1)2+3 ∣∣​20​13​∣∣​⇒C23​=−(6−0)=−6C31​=(−1)3+1 ∣∣​1−2​1−1​∣∣​⇒C31​=−1+2=1C32​=(−1)3+2 ∣∣​21​1−1​∣∣​⇒C32​=−(−3−0)=3C33​=(−1)3+3 ∣∣​21​1−2​∣∣​⇒C33​=−4−1=−5Hence, the co-factor matrix is C=⎣⎡​1381​5−103​3−6−5​⎦⎤​⇒adjA=CT=⎣⎡​1353​8−10−6​13−5​⎦⎤​⇒A−1=∣A∣adjA​=341​⎣⎡​1353​8−10−6​13−5​⎦⎤​Solution is given by ⎣⎡​xyz​⎦⎤​=341​⎣⎡​1353​8−10−6​13−5​⎦⎤​⎣⎡​13/29​⎦⎤​⎣⎡​xyz​⎦⎤​=341​⎣⎡​13+12+95−15+273−9−45​⎦⎤​⎣⎡​xyz​⎦⎤​=341​⎣⎡​3417−51​⎦⎤​⎣⎡​xyz​⎦⎤​=⎣⎡​11/2−3/2​⎦⎤​Hence, x=1,y=21​,z=−23​

Question Text	Solve the system of equations, using matrix method $2 x + y + z = 1, x - 2 y - z = \frac{3}{2}, 3 y - 5 z = 9$
Answer Type	Text solution:1
Upvotes	150

Solve the system of equations, using matrix method $2 x + y + z = 1, x - 2 y - z = \frac{3}{2}, 3 y - 5 z = 9$

Text solutionVerified

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Solve the system of equations, using matrix method2x+y+z=1, x−2y−z=23​, 3y−5z=9

Text solutionVerified

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Solve the system of equations, using matrix method $2 x + y + z = 1, x - 2 y - z = \frac{3}{2}, 3 y - 5 z = 9$