Prove that (AB)^-1 = B^-1 A^-1

This problem can be solved in 8 steps:

1. Let AB = C

2. A^-1AB = A^-1C

3. IB = A^-1C as the identity matrix I = A^-1A

4. B^-1B = B^-1A^-1C premultiply both sides by B^-1

5. I = B^-1A^-1C as B^-1B = I, the identity matrix

6. C^-1=B^-1A^-1CC^-1 post multiple both sides by C^-1

7. C^-1=B^-1A^-1 as CC^-1 = I, the identity matrix

8. (AB)^-1=B^-1A^-1