question_answer 1)

Let verify that       (i) [adj A]?1 = adj (A?1)       (ii) (A?1)?1 = A  

Answer:

      = 1(15?1) + 2 (?10?1) + 1(?2?3)       = 14 ? 22 ? 5 = ?13 0       Therefore A is non-singular matrix so A?1       exists.       Here       A11 = 14; B12 = 11; B13 = ?5       A21 = 11; B22 = 4; B23 = ?3        A31 = ?5; B32 = ?3; B33 = ?1                   Now             Let adj A = B             To find B?1             = 14 (?4 ?9) ? 11 (?11 ? 15) ? 5 (?33 + 20)       = ?182 + 286 + 65 = 169 0       Therefore B is non-singular matrix so B?1 exists.       A11 = ?13; B12 = 26; B13 = ?13       A21 = 26; B22 = ?39; B23 = ?13        A31 = ?13; B32 = 13; B33 = ?65       adj                          ?.. (1)       Also Let A?1 = C                                                 .. (2)       From (1) and (2), we get             [adj A]?1 = adj (A?1)        Part (i) is proved       Now                                                                    is also non-singular matrix so C?1 exists                                                                   Hence part (ii) is proved.