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Question
If A = `[(1,3,3),(1,4,3),(1,3,4)]` then verify that A(adj A) = |A| I and also find A-1.
Solution
Given A = `[(1,3,3),(1,4,3),(1,3,4)]`
|A| = `|(1,3,3),(1,4,3),(1,3,4)|`
`= 1|(4,3),(3,4)| - 3|(1,3),(1,4)| + 3|(1,4),(1,3)|`
= 1[16 – 9] – 3[4 – 3] + 3[3 – 4]
= 1(7) – 3(1) + 3(-1)
= 7 – 3 – 3
= 1
Cofactor [Aij] = `[(7,-1,-1),(-|(3,3),(3,4)|,|(1,3),(1,4)|,-|(1,3),(1,3)|),(|(3,3),(4,3)|,-|(1,3),(1,3)|,|(1,3),(1,4)|)]`
`= [(7,-1,-1),(-(12-9),(4-3),0),((9-12),0,(4-3))]`
`= [(7,-1,-1),(-3,1,0),(-3,0,1)]`
adj A = [Aij]T = `[(7,-3,-3),(-1,1,0),(-1,0,1)]`
Now A(adj A) = `[(1,3,3),(1,4,3),(1,3,4)][(7,-3,-3),(-1,1,0),(-1,0,1)]`
`= [(7-3-3,-3+3+0,-3+0+3),(7-4-3,-3+4,-3+0+3),(7-3-4,-3+3+0,-3+0+4)]`
`= [(1,0,0),(0,1,0),(0,0,1)]` ....(1)
|A| I = `1[(1,0,0),(0,1,0),(0,0,1)] = [(1,0,0),(0,1,0),(0,0,1)]` .....(2)
`"A"^-1 = 1/|"A"|` adj A
`= 1/1 [(7,-3,-3),(-1,1,0),(-1,0,1)] = [(7,-3,-3),(-1,1,0),(-1,0,1)]`
From (1) and (2),
A(Adj A) = |A| I
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