If A = and B = , verify that (AB)-1 = B-1 A-1.

Given:


To Verify: (AB)-1= B-1A-1


Firstly, we find the (AB)-1


Calculating AB





We have to find (AB)-1 and


Firstly, we find the adj AB and for that we have to find co-factors:


a11 (co – factor of -41) = (-1)1+1(26) = (-1)2(26) = 26


a12 (co – factor of 31) = (-1)1+2(-34) = (-1)3(-34) = 34


a21 (co – factor of -34) = (-1)2+1(31) = (-1)3(31) = -31


a22 (co – factor of 26) = (-1)2+2(-41) = (-1)4(-41) = -41



Now, adj AB = Transpose of co-factor Matrix



Calculating |AB|





= [-41 × 26 – (-34) × (31)]


= (-1066 + 1054)


= -12



Now, we have to find B-1A-1


Calculating B-1


Here,


We have to find A-1 and


Firstly, we find the adj B and for that we have to find co-factors:


a11 (co – factor of -4) = (-1)1+1(-4) = (-1)2(-4) = -4


a12 (co – factor of 3) = (-1)1+2(5) = (-1)3(5) = -5


a21 (co – factor of 5) = (-1)2+1(3) = (-1)3(3) = -3


a22 (co – factor of -4) = (-1)2+2(-4) = (-1)4(-4) = -4



Now, adj B = Transpose of co-factor Matrix



Calculating |B|





= [(-4) × (-4) – 3 × 5]


= (16 – 15)


= 1



Calculating A-1


Here,


We have to find A-1 and


Firstly, we find the adj A and for that we have to find co-factors:


a11 (co – factor of 9) = (-1)1+1(-2) = (-1)2(-2) = -2


a12 (co – factor of -1) = (-1)1+2(6) = (-1)3(6) = -6


a21 (co – factor of 6) = (-1)2+1(-1) = (-1)3(-1) = 1


a22 (co – factor of -2) = (-1)2+2(9) = (-1)4(9) = 9



Now, adj A = Transpose of co-factor Matrix



Calculating |A|





= [9 × (-2) – (-1) × 6]


= (-18 + 6)


= -12



Calculating B-1A-1


Here,


So,





So, we get


and


(AB)-1 = B-1A-1


Hence verified


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