Question

(15pt) Prove that (AB)^-1 = B^-1A^-1 provided A and B are invertible.

Answer 1

Answer:

See explanation below

Step-by-step explanation:

By definition of inverse we know that an operation multiplied by its inverse gives us 1.

Therefore we  can multiply (AB)⁻¹ by AB and the result will be 1

(AB)(AB)⁻¹ = 1

A⁻¹(AB)(AB)⁻¹= A⁻¹ (we multiplied both sides by A⁻¹ and on the left side now we have (A)(A)⁻¹= 1)

1·B(AB)⁻¹ = A⁻¹

B(AB)⁻¹ = A⁻¹

B⁻¹B(AB)⁻¹ = B⁻¹A⁻¹ (we multiplied both sides by B⁻¹ and on the left side we have B⁻¹B = 1)

1·(AB)⁻¹ = B⁻¹A⁻¹

Therefore, (AB)⁻¹ = B⁻¹A⁻¹