Solve the system of equations, using matrix method
$x - y + 2 z = 7, 3 x + 4 y - 5 z = - 5, 2 x - y + 3 z = 12$

Question

Solve the system of equations, using matrix methodx−y+2z=7, 3x+4y−5z=−5, 2x−y+3z=12

Filo · Accepted Answer

Given system of equationsx−y+2z=73x+4y−5z=−52x−y+3z=12This can be written as AX=Bwhere A=⎣⎡​132​−14−1​2−53​⎦⎤​,X=⎣⎡​xyz​⎦⎤​,B=⎣⎡​7−512​⎦⎤​Here, ∣A∣=1(12−5)+1(9+10)+2(−3−8)⇒∣A∣=7+19−22=4Since, ∣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 ∣∣​4−1​−53​∣∣​⇒C11​=12−5=7C12​=(−1)1+2 ∣∣​32​−53​∣∣​⇒C12​=−(9+10)=−19C13​=(−1)1+3 ∣∣​32​4−1​∣∣​⇒C13​=−3−8=−11C21​=(−1)2+1 ∣∣​−1−1​23​∣∣​⇒C21​=−(−3+2)=1C22​=(−1)2+2 ∣∣​12​23​∣∣​⇒C22​=3−4=−1C23​=(−1)2+3 ∣∣​12​−1−1​∣∣​⇒C23​=−(−1+2)=−1C31​=(−1)3+1 ∣∣​−14​2−5​∣∣​⇒C31​=5−8=−3C32​=(−1)3+2 ∣∣​13​2−5​∣∣​⇒C32​=−(−5−6)=11C33​=(−1)3+3 ∣∣​13​−14​∣∣​⇒C33​=4+3=7Hence, the co-factor matrix is C=⎣⎡​71−3​−19−111​−11−17​⎦⎤​⇒adjA=CT=⎣⎡​7−19−11​1−1−1​−3117​⎦⎤​⇒A−1=∣A∣adjA​=41​⎣⎡​7−19−11​1−1−1​−3117​⎦⎤​Solution is given by ⎣⎡​xyz​⎦⎤​=41​⎣⎡​7−19−11​1−1−1​−3117​⎦⎤​⎣⎡​7−512​⎦⎤​⎣⎡​xyz​⎦⎤​=41​⎣⎡​49−−5−36−133+5+132−77+5+84​⎦⎤​⎣⎡​xyz​⎦⎤​=41​⎣⎡​8412​⎦⎤​⎣⎡​xyz​⎦⎤​=⎣⎡​213​⎦⎤​Hence, x=2,y=1,z=3

Question Text	Solve the system of equations, using matrix method $x - y + 2 z = 7, 3 x + 4 y - 5 z = - 5, 2 x - y + 3 z = 12$
Answer Type	Text solution:1
Upvotes	150

Solve the system of equations, using matrix method
$x - y + 2 z = 7, 3 x + 4 y - 5 z = - 5, 2 x - y + 3 z = 12$

Text solutionVerified

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