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Question
Verify A (adj A) = (adj A) A = |A|I.
`[(1,-1,2),(3,0,-2),(1,0,3)]`
Solution
A ` = [(1,-1,2),(3,0,-2),(1,0,3)]`
|A| = 1[0 - 0] + 1[9 + 2] + 2[0 - 0]
= 1 × 11
= 11
`"A"_11 = (- 1)^(1 + 1) |(0,-2),(0,3)| = (- 1)^2 [0 - 0] = 0`
`"A"_12 = (- 1)^(1 + 2) |(3,-2),(1,3)| = (- 1)^3 [9 + 2] = (- 11) = - 11`
`"A"_13 = (- 1)^(1 + 3) |(3,0),(1,0)| = (- 1)^4 [0 - 0] = 0`
`"A"_21 = (- 1)^(2 + 1) |(-1,2),(0,3)| = (- 1)^3 [- 3 - 0] = - 1 xx (- 3) = 3`
`"A"_22 = (- 1)^(2 + 2) |(1, 2),(1,3)| = (- 1)^4 [3 - 2] = 1 xx 1 + 1`
`"A"_23 = (- 1)^(2+ 3) |(1,-1),(1,0)| = [0 - 1] = -1`
`"A"_31 = (- 1)^(3 + 1) |(-1,2),(0,-2)| = (- 1)^4 [2 - 0] = 1 xx 2 = 2`
`"A"_32 = (- 1)^(3 + 2) |(1,2),(3,-2)| = (- 1)^5 [- 2 - 6] = - 1 xx (- 8) = 8`
`"A"_33 = (- 1)^(3 + 3) |(1,-1),(3,0)| = (- 1)^6 [0 + 3] = 1 xx 3 = 3`
adj A = `[(0,-11,0),(3,1,-1),(2,8,3)] = [(0,3,2),(-11,1,8),(0,-1,3)]`
LHS = A
(adj A) `= [(1,-1,2),(3,0,-2),(1,0,3)] [(0,3,2),(-11,1,8),(0,-1,3)]`
`[(1 xx 0 + (- 1) xx (- 11) + 2 xx 0, 1 xx 3 + (- 1) xx 1 + 2 xx (- 1), 1 xx 2 + (- 1) xx 8 + 2 xx 3),(3 xx 0 + 0 xx (- 11) + (- 2) xx 0, 3 xx 3 + 0 xx 1 + (- 2) xx (- 1), 3 xx 2 + 0 xx 8 + (- 2) xx 3),(1 xx 0 + 0 xx (- 11) + 3 xx 0, 1 xx 3 + 0 xx 1 + 3 xx (-1), 1 xx 2 + 0 xx 8 + 3 + 3)]`
`= [(0+11+0,3 - 1 - 2, 2 - 8 + 6),(0+0+0, 9 + 0 + 2, 6 + 0 - 6),(0 + 0 + 0, 3 + 0 - 3, 2 + 0 + 9)]`
`= [(11,0,0),(0,11,0),(0,0,11)] = 11[(1,0,0),(0,1,0),(0,0,1)]` = 11 I = |A| · I
RHS = (adj A)A `= [(0,3,2),(-11,1,8),(0,-1,3)][(1,-1,2),(3,0,-2),(1,0,3)]`
`[(0xx3 + 3 xx 3 + 2 xx 1,0xx(-1) + 3 xx 0 + 2 xx 0,0 xx 2 + 3 xx (- 2) + 2 xx 3),(-11xx1 + 1 xx 3 + 8 xx 1, -11 xx (- 1) + 1 xx 0 + 8 xx 0,-11 xx 2 + 1 xx(- 2) + 8 xx3),(0 xx 1 + (- 1) xx 3 + 3 xx 1, 0xx(- 1) + (- 1) xx 0 + 3 xx 0, 0xx2 + (- 1)xx (- 2) + 3 xx 3)]`
`= [(0 + 9 + 2, 0 + 0 + 0, 0 - 6 + 6),(- 11 + 3 + 8, 11 + 0 + 0, - 22 - 2 + 24),(0 - 3 + 3, 0 + 0 + 0, 0 + 2 + 9)]`
`= [(11,0,0),(0,11,0),(0,0,11)] = 11. [(1,0,0),(0,1,0),(0,0,1)]` = 11 I = |A| I
det`A = | (1,-1,2),(3,0,-2), (1,0,3)|`
= 1 (0) + 1(9+2)+2(0)
= 11I
Hence, A(adj A) = (adj A) A = |A| I
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