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