Vtyjkyuk

Problem Set

I'm trying to solve this problem and I've managed to solve earlier parts that had to do with the rank depending on the λ and whether A can be inverted but I am not sure how to answer these three questions.

For the first question, I'm clueless, I've been searching for hours and haven't found something.

Then on the second one, I found some similar questions in here but they don't have a variable or that ^3. From what I've gathered so far, I can see that AAT is symmetric and therefore it has real eigenvalues. But I am not sure if λ affects it somehow or if that answer is good enough (suppose we didn't have that ^3) and then what to do with that ^3.

Finally on the third question I haven't spend that much time on it, because I feel like there is a chapter that has to do with AAT, ATA and so on that I might have missed that doesn't revolve around actually calculating (AATA)^5 and then seeing what happens to the determinant but somehow skipping all that with some properties I am not aware of.

I think that for the first problem the answer is that there are no such matrices, because:

If $A,B,C$ correspond to linear maps $a,b,c$ st $$a:mathbbR^3tomathbbR^3,quad b:mathbbR^3tomathbbR^4, quad c:mathbbR^4tomathbbR^3$$

then $BAC$ corresponds to $bcirc acirc c:mathbbR^4tomathbbR^4$ and you want that to be $1-1$ and onto. But that would mean that $b:mathbbR^3tomathbbR^4$ is onto which is a contradiction.