# Question #1d3d1

Feb 8, 2017

See below.

#### Explanation:

For part 1), let's say that $A = I$.

Well then $I + A {A}^{t} = 2 I$, which is invertible. So part 1) cannot be false for all $A$.

For part 2), if we say that $A = - I$, then $A {A}^{t} = \left(- I\right) {\left(- I\right)}^{t} = {\left(- 1\right)}^{t + 1} I$.

IOW, repeated eigenvalues, and for a 2x2, we have eigenvalues ${\lambda}_{1 , 2} = {\left(- 1\right)}^{t + 1}$.

So they're real and repeated, and of the same sign, BUT not always non-negative.

Those are the obvious counter-examples I could think of, but they're not that good, tbh.

But the question is so open you can let $A$ be the null matrix or any rotation matrix or a cyclic matrix or whatever you like.