MATH SOLVE

3 months ago

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

Accepted Solution

A:

Answer:See explanation belowStep-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 = 1A⁻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⁻¹Therefore, (AB)⁻¹ = B⁻¹A⁻¹