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Problem: inverse [[10,-9,-12],[7,-12,11],[-10,10,3]] [ Calculator, Method and examples ]

Solution:
Your problem -> inverse [[10,-9,-12],[7,-12,11],[-10,10,3]]

|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) =
 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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