A = ⎡⎢⎣2−3532−411−2⎤⎥⎦
|A|=2(−4+4)+3(−6+4)+5(3−2)=0−6+5=−1≠0
∴A is non-singular so its inverse exists.
Cij are the cofactors of the determinant,
C11=(−1)2 [2−41−2]=0
C12=(−1)3 [3−41−2]=2
C13=(−1)4 [3211]=1
C21=(−1)3 [−351−2]=−1
C22=(−1)4 [251−2]=−9
C23=(−1)5 [2−311]=−5
C31=(−1)4 [−352−4]=2
C32=(−1)5 [253−4]=23
C33=(−1)6 [2−332]=13
adj.A=⎡⎢⎣021−1−9−522313⎤⎥⎦T=⎡⎢⎣0−122−9231−513⎤⎥⎦
A−1=1|A|adj.A=1−1⎡⎢⎣0−122−9231−513⎤⎥⎦
= ⎡⎢⎣01−2−29−23−15−13⎤⎥⎦
Also, 2x−3y+5z=113x+2y−4z=−5 x+y−2z=−3
The given system of equations can be written as AX=B, where
A= ⎡⎢⎣2−3532−411−2⎤⎥⎦,X=⎡⎢⎣xyz⎤⎥⎦and B=⎡⎢⎣11−5−3⎤⎥⎦
So, X = A−1B
⎡⎢⎣xyz⎤⎥⎦=⎡⎢⎣01−2−29−23−15−13⎤⎥⎦×⎡⎢⎣11−5−3⎤⎥⎦
⎡⎢⎣xyz⎤⎥⎦=⎡⎢⎣ 0−5+6−22−45+69−11−25+39⎤⎥⎦=⎡⎢⎣123⎤⎥⎦
Hence, x=1,y=2 and z=3.