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प्रश्न
Verify A(adj A) = (adj A)A = |A|I.
`[(1,-1,2),(3,0,-2),(1,0,3)]`
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उत्तर
Let A = `[(1,-1,2),(3,0,-2),(1,0,3)]`
|A| = 1[0 − 0] + 1[9 + 2] + 2[0 − 0]
= 1 × 11
= 11
A11 = `(-1)^(1 + 1) |(0,-2),(0,3)|`
= (−1)2 [0 − 0]
= 0
A12 = `(-1)^(1 + 2) |(3,-2),(1,3)|`
= (−1)3 [9 + 2]
= −11
A13 = `(-1)^(1 + 3) |(3,0),(1,0)|`
= (−1)4 [0 − 0]
= 0
A21 = `(-1)^(2 + 1) |(-1,2),(0,3)|`
= (−1)3 [−3 − 0]
= −1 × (−3)
= 3
A22 = `(- 1)^(2 + 2) |(1, 2),(1,3)|`
= (−1)4 [3 − 2]
= 1 × 1
= 1
A23 = `(-1)^(2+ 3) |(1,-1),(1,0)|`
= (−1)5 [0 + 1]
= −1
A31 = `(1)^(3 + 1) |(-1,2),(0,-2)|`
= (−1)4 [2 − 0]
= 1 × 2
= 2
A32 = `(-1)^(3 + 2) |(1,2),(3,-2)|`
= (−1)5 [−2 − 6]
= −1 × (−8)
= 8
A33 = `(-1)^(3 + 3) |(1,-1),(3,0)|`
= (−1)6 [0 + 3]
= 1 × 3
= 3
adj A = `[(0,-11,0),(3,1,-1),(2,8,3)] = [(0,3,2),(-11,1,8),(0,-1,3)]`
L.H.S. = 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
R.H.S. = (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
Hence, A(adj A) = (adj A)A = |A|I
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