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Find A=[[10,-9,-12],[7,-12,11],[-10,10,3]], A^-1

Solution:
Your problem `->` A=[[10,-9,-12],[7,-12,11],[-10,10,3]], A^-1


`|A|` = 
 `10`  `-9`  `-12` 
 `7`  `-12`  `11` 
 `-10`  `10`  `3` 


 =
 `10` × 
 `-12`  `11` 
 `10`  `3` 
 `+9` × 
 `7`  `11` 
 `-10`  `3` 
 `-12` × 
 `7`  `-12` 
 `-10`  `10` 


`=10 xx (-12 × 3 - 11 × 10) +9 xx (7 × 3 - 11 × (-10)) -12 xx (7 × 10 - (-12) × (-10))`

`=10 xx (-36 -110) +9 xx (21 +110) -12 xx (70 -120)`

`=10 xx (-146) +9 xx (131) -12 xx (-50)`

`= -1460 +1179 +600`

`=319`


`Adj(A)` = 
Adj
`10``-9``-12`
`7``-12``11`
`-10``10``3`


 = 
 + 
 `-12`  `11` 
 `10`  `3` 
 - 
 `7`  `11` 
 `-10`  `3` 
 + 
 `7`  `-12` 
 `-10`  `10` 
 - 
 `-9`  `-12` 
 `10`  `3` 
 + 
 `10`  `-12` 
 `-10`  `3` 
 - 
 `10`  `-9` 
 `-10`  `10` 
 + 
 `-9`  `-12` 
 `-12`  `11` 
 - 
 `10`  `-12` 
 `7`  `11` 
 + 
 `10`  `-9` 
 `7`  `-12` 
T


 = 
`+(-12 × 3 - 11 × 10)``-(7 × 3 - 11 × (-10))``+(7 × 10 - (-12) × (-10))`
`-(-9 × 3 - (-12) × 10)``+(10 × 3 - (-12) × (-10))``-(10 × 10 - (-9) × (-10))`
`+(-9 × 11 - (-12) × (-12))``-(10 × 11 - (-12) × 7)``+(10 × (-12) - (-9) × 7)`
T


 = 
`+(-36 -110)``-(21 +110)``+(70 -120)`
`-(-27 +120)``+(30 -120)``-(100 -90)`
`+(-99 -144)``-(110 +84)``+(-120 +63)`
T


 = 
`-146``-131``-50`
`-93``-90``-10`
`-243``-194``-57`
T


 = 
`-146``-93``-243`
`-131``-90``-194`
`-50``-10``-57`


`"Now, "A^(-1)=1/|A| × Adj(A)`

 = `1/(319)` ×
`-146``-93``-243`
`-131``-90``-194`
`-50``-10``-57`


 = 
`-146/319``-93/319``-243/319`
`-131/319``-90/319``-194/319`
`-50/319``-10/319``-57/319`







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