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Question
Solve the following system of linear equations by matrix method.
x−y+2z=7
3x+4y−5z=−5
2x−y+3z=12.
x−y+2z=7
3x+4y−5z=−5
2x−y+3z=12.
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Solution
Let A=⎡⎢⎣1−1234−52−13⎤⎥⎦B=⎡⎢⎣7−512⎤⎥⎦X=⎡⎢⎣xyz⎤⎥⎦
Find |A|
|A|=∣∣
∣∣1−1234−52−13∣∣
∣∣=1(12−5)+1(9+10)+2(−3−8)
=7+19−22=26−22=4≠0
A exists
To find cofactors of A
A11=+∣∣∣4−5−13∣∣∣=7 A21=−∣∣∣−12−13∣∣∣=+1 A31=+∣∣∣−124−5∣∣∣=−3
A12=−∣∣∣3523∣∣∣=−19 A22=+∣∣∣1223∣∣∣=−1 A32=−∣∣∣1235∣∣∣=+11
A13=+∣∣∣342−1∣∣∣=−11 A23=−∣∣∣1−12−1∣∣∣=−1 A33=+∣∣∣1−134∣∣∣=7
∴Cofactor matrix=⎡⎢⎣7−19−111−103117⎤⎥⎦
adj A=⎡⎢⎣7−19−111−1−1−3117⎤⎥⎦=⎡⎢⎣71−3−19−111−11−17⎤⎥⎦
∴A=adj A|A|=14⎡⎢⎣713−19−111−11−17⎤⎥⎦
∴X=A B=14⎡⎢⎣713−19−111−11−17⎤⎥⎦⎡⎢⎣7−512⎤⎥⎦=14⎡⎢⎣49−5−36−133+5+132−77+5+84⎤⎥⎦=14⎡⎢⎣8412⎤⎥⎦
∴X=[1]⇒x=2;y=1;z=3.
Find |A|
|A|=∣∣
∣∣1−1234−52−13∣∣
∣∣=1(12−5)+1(9+10)+2(−3−8)
=7+19−22=26−22=4≠0
A exists
To find cofactors of A
A11=+∣∣∣4−5−13∣∣∣=7 A21=−∣∣∣−12−13∣∣∣=+1 A31=+∣∣∣−124−5∣∣∣=−3
A12=−∣∣∣3523∣∣∣=−19 A22=+∣∣∣1223∣∣∣=−1 A32=−∣∣∣1235∣∣∣=+11
A13=+∣∣∣342−1∣∣∣=−11 A23=−∣∣∣1−12−1∣∣∣=−1 A33=+∣∣∣1−134∣∣∣=7
∴Cofactor matrix=⎡⎢⎣7−19−111−103117⎤⎥⎦
adj A=⎡⎢⎣7−19−111−1−1−3117⎤⎥⎦=⎡⎢⎣71−3−19−111−11−17⎤⎥⎦
∴A=adj A|A|=14⎡⎢⎣713−19−111−11−17⎤⎥⎦
∴X=A B=14⎡⎢⎣713−19−111−11−17⎤⎥⎦⎡⎢⎣7−512⎤⎥⎦=14⎡⎢⎣49−5−36−133+5+132−77+5+84⎤⎥⎦=14⎡⎢⎣8412⎤⎥⎦
∴X=[1]⇒x=2;y=1;z=3.
A exists
To find cofactors of A
A11=+∣∣∣4−5−13∣∣∣=7 A21=−∣∣∣−12−13∣∣∣=+1 A31=+∣∣∣−124−5∣∣∣=−3
A12=−∣∣∣3523∣∣∣=−19 A22=+∣∣∣1223∣∣∣=−1 A32=−∣∣∣1235∣∣∣=+11
A13=+∣∣∣342−1∣∣∣=−11 A23=−∣∣∣1−12−1∣∣∣=−1 A33=+∣∣∣1−134∣∣∣=7
∴Cofactor matrix=⎡⎢⎣7−19−111−103117⎤⎥⎦
adj A=⎡⎢⎣7−19−111−1−1−3117⎤⎥⎦=⎡⎢⎣71−3−19−111−11−17⎤⎥⎦
∴A=adj A|A|=14⎡⎢⎣713−19−111−11−17⎤⎥⎦
∴X=A B=14⎡⎢⎣713−19−111−11−17⎤⎥⎦⎡⎢⎣7−512⎤⎥⎦=14⎡⎢⎣49−5−36−133+5+132−77+5+84⎤⎥⎦=14⎡⎢⎣8412⎤⎥⎦
∴X=[1]⇒x=2;y=1;z=3.
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