Prove the following. In each case clearly state in mathematical notation what is given and what you are asked to prove.

a) Given that A and B are square matrices that commute and satisfies

A2 − AB − 2B2 − I = 0

Prove that the inverse of (A+B) exists and find it.

Hint: assume the inverse is of the form (αA + βB) or some constants α and β.

b) Given that A , B and C are symmetric square matrices of the same size, prove that the matrix AT + B + CT is symmetric.

AND...

What is the rank of an invertible 5x5 matrix? Why?

Thanks a lot. Any suggestions would be appreciated.