आव्यूह A का परिवर्त तथा सहखंडज ज्ञात करे , जहाँ
`A= [(1,2,3),(0,5,4),(3,0,3)]`
(Also verify that ) A (adj A) = |A| I को भी सत्यापित करे ।

Explanation

माना कि `A= [(1,2,3),(0,5,0),(2,4,3)]= [(a_(11) ,a_(12),a_(13)),(a_(21),a_(22),a_(23)),(a_(31),a_(32),a_(33))]`
` A_(1) = a_(11) ` अर्थात 1 का cofactor = `= (1)^(1+1) |(5,0),(5,3)| = 15 - 0 = 15`
` A_(12) = a_(12) ` अर्थात 2 का cofactor ` = (-1)^(1+3) |(0,0),(2,3)| = - (0,0) = 0 `
` A_(13) = a_(13) `अर्थात 3 का cofactor ` = (-1)^(1+3) |(0,5),(2,4)| =0 - 10 = - 10 `
` A_(21) = a_(21) ` अर्थात 0 का cofactor ` = (-1)^(2+1) |(2,3),(4,3) | = - (6 - 12) = 6`
` A_(22) = a_(22)` अर्थात 5 का cofactor ` = (-1)^(2+2) |(1,3),(2,3)| = 3 - 6 = - 3 `
`A_(23) = a_(23) ` अर्थात 0 का cofactor ` = (-1)^(2+3) |(1,2),(2,4)| =- (4-4) = 0 `
` A_(31) = a_(31) ` का cofactor` = (-1)^(3+1) |(2,3),(5,0)| = (0-15) = - 15`
` A_(32) = a_(32) ` का cofactor ` = (-1)^(3+2) |(1,3),(0,0) = 0 `
` A_(33) = a_(33) ` का cofactor `= (-1)^(3+3) |(1,2),(0,5)| = 5- 0 = 5`
माना कि `B= [(A_(11) ,A_(12),A_(13)),(A_(21),A_(22),A_(23)),(A_(31),A_(32),A_(33))]= [(15,0,-10),(6,-3,0),(-15,0,5)]`
` therefore A=B'|(15,6,-15),(0,-3,0),(-10,0,5)|`
Second Part : `|A| = 1 (15-0)-2(0-0) + 3 (0-10) = - 15`
अब A(adj A) =` [(1,2,3),(0,5,0),(2,4,3)][(15,6,-15),(0,-3,0),(-10,0,5)]`
= `[(15+0-30,6-6+0,-15+0+15),(0+0+0, 0-15+0,0+0+0),(30 + 0- 30 , 12 - 12 + 0 , - 30 + 0 + 15)]`
` [(-15,0,0),(0,-15,0),(0,0,-15)]=-15[(1,0,0),(0,1,0),(0,0,1)] = - 15L = |A|I`
अतः A(adi A) = |A|I.