Examine the consistency of the system of equations
`3x -y - 2z = 2" "``2y -z = 1" "``3x - 5y = 3`

Explanation

The system of equations can be written as

`3x -y - 2z = 2`

`0x+2y -z = 1`

`3x - 5y+0z = 3`

Writing equation as`AX=B`

`[[3,-1,-2],[0,2,-1],[3,-5,0]][[x],[y],[z]]=[[2],[1],[3]]`

Here,
`A=[[3,-1,-2],[0,2,-1],[3,-5,0]]`, `X=[[x],[y],[z]]` & `B=[[2],[1],[3]]`


Calculating `|A|`

`|A|=|[3,-1,-2],[0,2,-1],[3,-5,0]|`

` " " " "=3|[2,-1],[-5,0]|-(-1)|[0,-1],[3,0]|-2|[0,2],[3,-5]|`

` " " " "=3(0-5)+1(0+3)-2(0-6)=3(-5)+1(3)-2(-6)`

` " " " "=-15+3+12=-15+15`

`|A|=0`


Calculating `adjA(B)`

`adjA=[[A_11,A_12,A_13],[A_21,A_22,A_23],[A_31,A_32,A_33]]^'=[[A_11,A_21,A_31],[A_12,A_22,A_32],[A_13,A_23,A_33]]`

`A=[[3,-1,-2],[0,2,-1],[3,-5,0]]`


`M_11=|[2,-1],[-5,0]|=0-5=-5`, ` " " " "` `M_12=|[0,-1],[3,0]|=0+3=3`, ` " " " "` `M_13=|[0,2],[3,-5]|=0-6=-6`

`M_21=|[-1,-2],[-5,0]|=0-10=-10`,` " " " "` `M_22=|[3,-2],[3,0]|=0+6=6`, ` " " " "` `M_23=|[3,-1],[3,-5]|=-15+3=-12`

`M_31=|[-1,-2],[2,-1]|=1+4=5`, ` " " " "` `M_32=|[3,-2],[0,-1]|=-3+0=-3`` " " " "` `M_33=|[3,-1],[0,2]|=6+0=6`

Now,
`A_11=(-1)^(1+1).M_11=(-1)^2.(-5)=-5`` " " " "` `A_12=(-1)^(1+2).M_12=(-1)^3=-3`` " " " "``A_13=(-1)^(1+3).M_13=(-1)^4.(-6)=-6`

`A_21=(-1)^(2+1).M_21=(-1)^3.(-10)=10` ` " " " "``A_22=(-1)^(2+2).M_22=(-1)^4.(6)=6` ` " " " "` `A_23=(-1)^(2+3).M_23=(-1)^5.(-12)=12`

`A_31=(-1)^(3+1).M_31=(-1)^4.(5)=5` ` " " " "` `A_32=(-1)^(3+2).M_32=(-1)^5.(-3)=3`` " " " "``A_33=(-1)^(3+3).M_33=(-1)^6.(6)=6`

Thus, `adj(A)=[[-5,10,5],[-3,6,3],[-6,12,6]]`

Now,

`adj(A).B`
Putting values

` " " " "=[[-5,10,5],[-3,6,3],[-6,12,6]][[2],[1],[3]]`

` " " " "=[[-5(2)+10(1)+5(3)],[-3(2)+6(1)+3(3)],[-6(2)+12(1)+6(3)]]=[[-10+10+15],[-6+6+9],[-12+12+18]]=[[15],[9],[18]]`

Thus, `adj A.B ne O`

Since
'|A|=0' & `(adj A)B ne O`

Thus, the given system equation is inconsistent & the system of equations has no solution.