Let A=[3725] and B=[6879] verify that
(AB)−1=B−1A−1
Let A=[3725] and B=[6879] verify that
(AB)−1=B−1A−1
Given, A=[3725]|A|=[3725]=15−14=1
Cofactors of A are A11=5,A12=−2,A21=−7,A22=3adj (A)=[5−2−73]T=[5−7−23]
Now, A−1=1|A|(adj(A))=11[5−7−23]
Here, B=[6879] ∴ |B|=[6879]=54−56=−2
Cofactors of B are B11=9,B12=−7,B21=−8,B22=6
adj (A)=[9−7−86]T=[9−8−76] ∴B−11|B|(adj B)=1−2[9−8−76]
Now, B−1A−1=1−2[9−8−76][5−7−23]=1−2[45+16−63−24−35−1249+18]=1−2[61−87−4767]=⎡⎣−612872472−672⎤⎦ …(i)
Now, AB=[3725][6879]=[18+4924+6312+3516+45]=[67874761]
∴ |AB|=[67874761]=67×61−47×87=4087−4089=−2
Cofactors of AB are A11=61,A12=−47,A21=−87,A22=67
adj(AB)=[61−47−8767]=[61−87−4767]
∴ (AB)−1=1|AB|(adj AB)=1−2[67−87−4767]=⎡⎣−612872472−672⎤⎦ …(ii)
From Eqs. (i) and (ii), we get (AB)−1=B−1A−1
Hence, the given result is proved.
Given, A=[3725]|A|=[3725]=15−14=1
Cofactors of A are A11=5,A12=−2,A21=−7,A22=3adj (A)=[5−2−73]T=[5−7−23]
Now, A−1=1|A|(adj(A))=11[5−7−23]
Here, B=[6879] ∴ |B|=[6879]=54−56=−2
Cofactors of B are B11=9,B12=−7,B21=−8,B22=6
adj (A)=[9−7−86]T=[9−8−76] ∴B−11|B|(adj B)=1−2[9−8−76]
Now, B−1A−1=1−2[9−8−76][5−7−23]=1−2[45+16−63−24−35−1249+18]=1−2[61−87−4767]=⎡⎣−612872472−672⎤⎦ …(i)
Now, AB=[3725][6879]=[18+4924+6312+3516+45]=[67874761]
∴ |AB|=[67874761]=67×61−47×87=4087−4089=−2
Cofactors of AB are A11=61,A12=−47,A21=−87,A22=67
adj(AB)=[61−47−8767]=[61−87−4767]
∴ (AB)−1=1|AB|(adj AB)=1−2[67−87−4767]=⎡⎣−612872472−672⎤⎦ …(ii)
From Eqs. (i) and (ii), we get (AB)−1=B−1A−1
Hence, the given result is proved.
