निम्नलिखित आव्यूह का प्रतिलोम निकाले ।
`[(0,1,2),(1,2,3),(3,1,1)]`

Explanation

माना कि `A= [(0,1,2),(1,2,3),(3,1,1)]= [(a_(11) ,a_(12),a_(13)),(a_(21),a_(22),a_(23)),(a_(31),a_(32),a_(33))]`
अब `।A।= [(0,1,2),(1,2,3),(3,1,1)]= 0 (2-3)- (1-9)+ 2 (1-6) = - 2 ne 0 `
अब `A^(-1)` का अस्तित्व है ।
अब `A_(11) = a_(11) ` अर्थात 0 का cofactor = `= (-1)^(1+1) |(2,3),(1,1)| = 2-3 = - 1`
`A_(12) = a_(12) ` अर्थात 0 का cofactor ` = (-1)^(1+2) |(1,3),(3,1)| = - (1-9) = 8`
`A_(13) = a_(13) ` अर्थात 2 का cofactor `= (-1)^(1+3) |(1 ,2),(3,1)| = 1-6 = - 5`
` A_(21) = a_(21)` अर्थात 1 का cofactor ` = (-1)^(2+1) |(0,2),(1,1)| = - (1-2)=1`
` A_(22) = a_(22)` अर्थात 2 का cofactor = ` (-1)^(2+2) |(0,1),(3,1)| = 0-6= - 6`
`A_(23)= a_(23) ` अर्थात 3 का cofactor ` = (-1)^(2+3) |(0,1),(3,1)| = - (0- 3) = 3`
`A_(31) = a_(31)` अर्थात 3का cofactor ` = (-1)^(3+2) |(1,2),(2,3)|= 3 - 4 = - 1`
` A_(32) = a_(32)` अर्थात 1 का cofactor ` = (-1)^(3+2) |(0,2),(1,3)| = - (0-2) = 2`
`A_(33) = a_(33) ` अर्थात 1 का cofactor ` = (-1)^(3+3) |(0,1),(1,2)| = 0-1= -1`
माना कि ` [(A_(11) ,A_(12),A_(13)),(A_(21),A_(22),A_(23)),(A_(31),A_(32),A_(33))]= |(-1,1,-5),(1-,6,3),(-1,2,-1)|`
` therefore "adj A = B'= [(-1,1,-1),(8,-6,2),(-5,3,-1)]`
`therefore A^(-1) = ("adj A")/(|A|) = ("adj A")/(-2) = [((1)/(2) ,-(1)/(2),(1)/(2)),(-4,3,-1),((5)/(2),-(3)/(2),(1)/(2))]`.