1.

If `A=[{:(3,2,1),(0,-1,-2),(-3,4,2):}]`, find `A^(-1)`

Answer» `|A|=[{:(3,2,1),(0,-1,-2),(-3,4,2):}]`
`=3[{:(-1,-2),(4,2):}]-2|{:(0,-2),(-3,2):}|+1|{:(0,-1),(-3,4):}|`
`=3(-2+8)-2(0-6)+1(0-3)
=18+12-3=27ne0`
`therefore" A is invertible matrix".`
Cofactore of matrix A
`C_(11)=(-1)^(1+1)|{:(-1,-2),(4,2):}|=(-2+8)=6`
`C_(12)=(-1)^(1+2)|{:(0,-2),(-3,2):}|=-(0-6)=6`
`C_(13)=(-1)^(1+3)|{:(0,-1),(-3,4):}|=0-3=-3`
`C_(21)=(-1)^(2+1)|{:(2,1),(4,2):}|=-(4-4)=0`
`C_(22)=(-1)^(2+2)|{:(3,1),(-3,2):}|=6+3=6`
`C_(23)=(-1)^(2+3)|{:(3,2),(-3,4):}|=-(12+6)=-18`
`C_(31)=(-1)^(3+1)|{:(2,1),(-1,-2):}|=-4+1=-3`
`C_(32)=(-1)^(3+2)|{:(3,2),(0,-2):}|=-(-0)=6`
`C_(33)=(-1)^(3+3)|{:(3,2),(0,-1):}|=-3-0=-3`
`therefore" adj.A"=[{:(c_(11),c_(12),c_(13)),(c_(21),c_(22),c_(23)),(c_(31),c_(32),c_(33)):}]`
`=[{:(6,6,-3),(0,9,-18),(-3,6,-3):}]=[{:(6,0,-3),(6,9,6),(-3,-18,-3):}]`
`"and "A^(-1)=1/(|A|)."adj.A"=1/27[{:(6,0,-3),(6,9,6),(-3,-18,-3):}]`
`=3/27[{:(2,0,-1),(2,3,2),(-1,-6,-1):}]=1/9[{:(2,0,-1),(2,3,2),(-1,-6,-1):}]`


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