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Question
Find the inverse of the matrices (if it exists).
`[(2,1,3),(4,-1,0),(-7,2,1)]`
Solution
A = `[(2,1,3),(4,-1,0),(-7,2,1)]`
So, adj A `= [(A_11,A_21,A_31),(A_12,A_22,A_32),(A_13,A_23,A_33)]`
`= [(-1,5,3),(-4,23,12),(1,-11,-6)]`
`A^-1` = 2(-1 - 0) - 1(4 - 0) + 3 (8 - 7)
`= -3 ne 0 -> A^-1`
`C_11 = (-1)^(1+1) |(1,0), (2,1)| = -1`
`C12 = (-1)^(1+2) |(4,0), (-7,1)| = -4`
`C_13 = (-1)^(1+3)|(4,-1),(-7,2)| =8 - 7 = 1`
`C_21 = (-1)^(2+1) |(1,3), (2,1)| = -(1-6) = 5`
`C_22 = (-1)^(2+2) |(2,3), (-7,1)| = 2+21 = 23`
`C_23 = (-1)^(2+3) |(2,1), (-7,2)| = -(4+7) = -11`
`C_31 = (-1)^(3+1) |(1,3), (-1,0)| = 2`
`C_32 = (-1)^(3+2) |(2,3), (4,0)| = 12`
`C_33 = (-1)^(3+3)|(2,1), (4,-1)| = -6`
`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/-3 [(-1,5,3),(-4,23,12),(1,-11,-6)]`
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