माना कि A = विकर्ण [a ,b , c ] जहाँ , a ,b , c शून्येतर है । `A^(-1)` ज्ञात करे ।

Explanation

दिया है `A = "diag [a,b,c] " therefore A = ।(a ,0,0),(0,b,0),(0,0,c)।`
अब `। A। = ।(a ,0,0),(0,b,0),(0,0,c)। = abc ne 0 " " [ because a ne 0 , b ne 0 , c ne 0]`
अतः `A^(-1)` का अस्तित्व है ।
अब `A_(11) = a` का cofactor `= (-1)^(1+1) |(b,0),(0,c)| = bc `
` A_(12) = 0 ` का cofactor ` = (-1)^(1+2) |(0,0),(0,c)| = 0 `
इसी तरह ` A_(13) = 0, A_(21) = 0 , A_(22) = ac , A_(23) = 0 , A_(31) = 0, A_(32) = 0 , A_(33) = ab `
`therefore B = [(A_(11) , A_(12), A_(13)),(A_(21),A_(22),A_(23)),(A_(31),A_(32), A_(33))]= [(bc,0,0),(0,ac,0),(0,0,ab)]`
`therefore "adj A = B" [(bc,0,0),(0,ac,0),(0,0,ab)]`
`therefore A^(-1) = ("adj A")/(|A|) = (1)/(abc)[(bc,0,0),(0,ac,0),(0,0,ab)]= [((1)/(a) ,0,0),(0,(1)/(b),0),(0,0,(1)/(c))]`
`therefore A^(-1) = "diag" [(1)/(a)(1)/(b)(1)/(c)] = "diag" [ a^(-1) b^(-1) c^(-1)]`.