I'm trying to prove two statements related to matrices but I can't find a good way to prove it: $$ T:R^{2}\rightarrow R^{2} $$$$ S:R^{2}\rightarrow R^{2} $$ Are linear transformations. What is the best way to prove that, if T and S are nondiagonalizable then (T ◦ S) is nondiagonalizable ?
I'm also trying to prove that, If T and S are non-invertible then (T + S) is non-invertible.
Those two questions are giving me some trouble - I can't find a good way to prove it.
Thanks a lot!
The second one is false, because let $A:= \begin{pmatrix} 1&0\\0&0 \end{pmatrix}$ and $B:=\begin{pmatrix} 0&0\\0&1 \end{pmatrix}$. A and B have rank 1 so A and B aren't invertible. But $A+B=I_2$ wich is invertible.