Question

Prove that (AB)−1=B−1A−1

Answer 1

Answer with Step-by-step explanation:

Consider,

`(AB)^(-1)(AB)=I` (Identity rule)

Multiplying by B⁻¹ on the both the sides, we get that

`(AB)^(-1)(AB)B^(-1)=IB^(-1)\\\\(AB)^(-1)A(BB^(-1))=B^(-1)`

And we know that BB⁻¹ = I

So, it becomes,

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

Now, multiplying by A⁻¹ on both the sides, we get that

`(AB)^(-1)AA^(-1)=B^(-1)A^(-1)\\\\(AB)^(-1)=B^(-1)A^(-1)` (AA⁻¹=I)

Hence, proved.