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Question
Find the inverse of the following matrix:
`[(1,2,3),(0,2,4),(0,0,5)]`
Solution
Let A = `[(1,2,3),(0,2,4),(0,0,5)]`
|A| = `|(1,2,3),(0,2,4),(0,0,5)|`
`= 1|(2,4),(0,5)| - 2|(0,4),(0,5)| + 3|(0,2),(0,0)|`
= 1[10 - 0] - 2[0 - 0] + 3[0 - 0]
= 10 - 0 + 0
= 10
[Aij] = `[(10,0,0),(-|(2,3),(0,5)|,|(1,3),(0,5)|,-|(1,2),(0,0)|),(|(2,3),(2,4)|,-|(1,3),(0,4)|,|(1,2),(0,2)|)]`
`= [(10,0,0),(-(10 - 0),(5-0),0),(8-6,-(4-0),2)]`
`= [(10,0,0),(-10,5,0),(2,-4,2)]`
adj A = [Aij]T = `[(10,-10,2),(0,5,-4),(0,0,2)]`
`"A"^-1 = 1/|"A"|`adj A
`= 1/10[(10,-10,2),(0,5,-4),(0,0,2)]`
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Solution:
AB = `[(4, 3, 2),(-1, 2, 0)] [(1, 2),(-1, 0),(1, -2)]`
AB = [ ]
|AB| = `square`
M11 = –2 ∴ A11 = (–1)1+1 . (–2) = –2
M12 = –3 A12 = (–1)1+2 . (–3) = 3
M21 = 4 A21 = (–1)2+1 . (4) = –4
M22 = 3 A22 = (–1)2+2 . (3) = 3
Cofactor Matrix [Aij] = `[(-2, 3),(-4, 3)]`
adj (A) = [ ]
A–1 = `1/|A| . adj(A)`
A–1 = `square`
