MathDB

Given positive integers

m,n

and a

m\times n

matrix

A

with real entries. (1) Show that matrices

X = I_m + AA^T

and

Y = I_n + A^T A

are invertible. (

I_l

is the

l\times l

unit matrix.) (2) Evaluate the value of

\text{tr}(X^{-1}) - \text{tr}(Y^{-1})

3