Let `A=[{:(1,-2,1),(-2,3,1),(1,1,5):}]`. Verify that ltbtgt (i) `[adjA]^(-1)=adj (A^(-1))`
(ii) `(A^(-1)^(-1)=A`

Explanation

`A=[{:(1,-2,1),(-2,3,1),(1,1,5):}]`
`therefore" "|A|=[{:(1,-2,1),(-2,3,1),(1,1,5):}]`
`=1(15-1)-(-2)(-10-1)+1(-2-3) `
`=14-22-5-13ne0`
`A_(11)=14, A_(12)=11, A_(13)=-5`
`A_(21)=11, A_(22)=4, A_(23)=-3`
`A_(31)=-5, A_(32)=-3, A_(33)=-1`
`therefore" "A=[{:(14,11,-5),(11,4,-3),(-5,-3,-1):}]=[{:(14,11,-5),(11,4,-3),(-5,-3,-1):}]`
`"and A"^(-1)=1/|A|="adj A"=-1/13[{:(14,11,-5),(11,4,-3),(-5,-3,-1):}]=[{:(-14/13,-11/13,5/13),(122/12,-4/13,3/13),(5/13,3/13,1/13):}]`
(i) `Let B = adj A=[{:(14,11,-5),(11,4,-3),(-5,-3,-1):}]rArr|B|=[{:(14,11,-5),(11,4,-3),(-5,-3,-1):}]`
=14(-4-9)=11(-11-15)-5(-33+20)
`=-182+286+65=169ne0`
`B_(11)=-13, B_(12)=26, B_(13)=-13`
`B_(21)=26, B_(22)=-39, B_(23)=-13`
`B_(31)=-13, B_(32)=-13, B_(33)=-65`
`therefore" adj B"=[{:(-13,26,-13),(26,-39,-13),(-13,-13,-65):}]=[{:(-13,26,-13),(26,-39,-13),(-13,-13,-65):}]`
`"andB"^(-1)=1/|B|"adj B="1/169[{:(-13,26,-13),(26,-39,-13),(-13,-13,-65):}]=1/13[{:(-1,2,-1),(1,-3,-1),(-1,-1,-5):}]`
`"Let C"=A^(-1)=[{:(-14/13,-11/13,5/13),(-11/13,-4/13,3/13),(5/13,3/13,1/13):}]`
`C_(11)=-13/169=-1/13,C_(12)=26/169=2/13, C_(13)=-13/169=-1/13`
`C_(21)=26/169=2/13, C_(22)=39/169=-3/13, C_(23)=-13/169=-1/13`
`C_(31)=-13/169=1/13, C_(32)=-13/169=-1/13, C_(33)=-65/169=-5/13`
`:."adjC=adjA"^(-1)=[{:(-1/13,2/13,-1/13),(2/13,-3/13,-1/13),(-1/13,-1/13,-5/13):}]=[{:(-1/13,2/13,-1/13),(2/13,-3/13,-1/13),(-1/13,-1/13,-5/13):}]`
`=1/13[{:(-1,2,-1),(2,-3,-1),(-1,-1,-5):}]`
(ii) `|A^(-1)|=14/13(=4/169-9/169)+11/13(-11/169-15/169)+5/13(-33/169+20/169)=1-1/13`
`therefore (A^(-1))^(-1)=1/|A^(-1)|"adj"(A^(-1))`
`1/(-1/13)-1/13[{:(-1,2,-1),(2,-3,-1),(-1,-1,-5):}]=[{:(1,-2,1),(2,3,1),(-1,1,5):}]=A`