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Question
Find the inverse of the matrices (if it exists).
`[(1,2,3),(0,2,4),(0,0,5)]`
Solution
A = `[(1,2,3),(0,2,4),(0,0,5)]`
`C_11 = (-1)^(1+1) |(2,4), (0,5)| = 10`
`C12 = (-1)^(1+2) |(0,4), (0,5)| = 0`
`C_13 = (-1)^(1+3)|(0,2),(0,0)| = 0`
`C_21 = (-1)^(2+1) |(2,3), (0,5)| = -10`
`C_22 = (-1)^(2+2) |(1,3), (0,5)| = 5`
`C_23 = (-1)^(2+3) |(1,2), (0,0)| = 0`
`C_31 = (-1)^(3+1) |(2,3), (2,4)| = 2`
`C_32 = (-1)^(3+2) |(1,3), (0,4)| = -4`
`C_33 = (-1)^(3+3)|(1,2), (0,2)| = 2`
So, adjA `= [("A"_11,"A"_21,"A"_31),("A"_12,"A"_22,"A"_32),("A"_13,"A"_23,"A"_33)]`
`= [(10,-10,2),(0,5,-4),(0,0,2)]`
`abs "A" = 1 (10 -0) - 2 (0 - 0) + 3 (0 - 0) = 10 ne 0 -> "A"^-1`
`"A"^-1 = 1/abs "A" ("adjA") = 1/abs "A" [("A"_11,"A"_21,"A"_31),("A"_12,"A"_22,"A"_32),("A"_13,"A"_23,"A"_33)]`
`= 1/10 [(10,-10,2),(0,5,-4),(0,0,2)]`
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