If two matrices $M$ and $N$ are related as $M^2 = N^2$ and they are not singular, can it be concluded that $M=N$?
If $M^2= N^2$ and $MN =NM$ also, and it is known that $M \ne N$, can we conclude they are singular?
I would really appreciate someone throwing light on the relationship between singularity and equality, along with a few examples.
No, If $M$ is any non-singular matrix and $N=-M$ you would have $N^2 = (-M)(-M) = M^2$, but $M\ne N$. With these matrices you also have $MN=M(-M)=-M^2 = (-M)M = NM$ so your second hypothesis is false too.
On the other hand if $M^2 = N^2$ and $MN = NM$ and $M\ne N$ you can conclude that $M+N$ is singular because $(M+N)(M-N) = M^2 + NM - MN - N^2$ and since $NM=MN$ this becomes $M^2-N^2$ (which is the null matrix according to assumption). If $M+N$ wasn't singular you could multiply the equation from the left by it's inverse and get $M-N = (M+N)^{-1}(M+N)(M-N) = M-N = (M+N)^{-1}0 = 0$ (which contradicts the assumption).