# Math Help - A couple of Matrix Algebra Problems

1. ## A couple of Matrix Algebra Problems

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.

2. Originally Posted by Nitz456
What is the rank of an invertible 5x5 matrix? Why?
the rank of an nxn matrix A is equal to the number of non-zero rows A has when it is put in reduced row-echelon form. since A is invertible, it means its reduced row echelon form is the identity matrix $I_n$. Thus, the reduced row-echelon form of a 5x5 matrix is $I_5$ which means the rank for such a matrix is 5