Question

Question #1d3d1

Answer 1

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 = (-I) (-I)^t = (-1)^(t+1) I`.

IOW, repeated eigenvalues, and for a 2x2, we have eigenvalues `lambda_(1,2) = (-1)^(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.