If `A=[(1,2,-3),(2,3,2),(3,-3,-4)]` then find `A^-1` and hence solve the follwoing equations: `x+2y-3z=4, 2x+3y+2z=2 and 3x-3y-4z=11`

If `A=[(1,2,-3),(2,3,2),(3,-3,-4)]` then find `A^-1` and hence solve t

1.	If `A=[(1,2,-3),(2,3,2),(3,-3,-4)]` then find `A^-1` and hence solve the follwoing equations: `x+2y-3z=4, 2x+3y+2z=2 and 3x-3y-4z=11`
Answer» The given equation are `x+2y-3z=-4. " "…(i)` `2x+3y+2z=2," "…(ii)` `3x-3y-4z=11." "…(iii)` `Let A=[{:(1 " "2 " "-3),(2 " "3 " " 2),(3 " -3 " -4):}],X=[{:(x),(y),(z):}] and B=[{:(-4),(2),(11):}]` So , the given system in matrix form is AX=B. `Now, \|A\|=[{:(1 " "2 " "-3),(2 " "3 " " 2),(3 " -3 " -4):}]=[{:(1 " "2 " "-3),(0 ""-1 " " 8),(0 " -9 " 5):}]` `[R_2rarrR_2-2R_1 and R_3rarrR_3-3R_1]=1.(-5+72)=67 ne 0` Thus , A is invertible So the has a unique solution , `X=A^(-1)B` Now , the confactors of the elements of \|A\| are `A_(11)=-6, A_(12)=14, A_(13)=-15` `A_(21)=17, A_(22)=5, A_(23)=9` `A_(31)=13, A_(32)=-8, A_(33)=-1` `therefore adjA=[{:(-6 " "14 " "-15),(17 " "5 " "9),(13 " "-8 " "-1):}]=[{:(-6 " "17 " "13),(14 " "5 " "-8),(-15 " "9 " "-1):}]` So,`A^(-1)=1/\|A\|.adjA=1/(67).[{:(-6 " "17 " "13),(14 " "5 " "-8),(-15 " "9 " "-1):}]` `thereforeX=A^(-1)B` or `[{:(x),(y),(z):}]=1/(67).[{:(-6 " "17 " "13),(14 " "5 " "-8),(-15 " "9 " "-1):}].[{:(-4),(2),(11):}]` `=1/(67).[{:(201),(-134),(67):}]=[{:(3),(-2),(1):}]` `therefore` x=3 , y=-2 and z=1