If $I≠T∈M_4(C)$ has $(x-1)^4 $ as it characteristics polynomial then what is the largest Dimension of the centralizer of T in$ M_4(C)$(= the subspace of all matrices that commute with T) ?
my answer :First i take T=$\left(\begin{matrix} 1 & 0 & 1 & 0\\ 0 & 1 & 0 & 1 \\ 0 & 0& 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{matrix}\right)as $$T≠I$
I think largest dimension will be 4 that is Identity matrix of 4×4 matrices
Is its correct or incorrect? Pliz tell me,,any hints will be appreciated
Verify that the dimension of the centralizer of $$ M = \pmatrix{1&1&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1} $$ is $6$. In particular, it will consist of all matrices of the form $$ \pmatrix{a_1&a_2\\&a_1\\&&b_{11}&b_{12}\\&&b_{21} & b_{22}} $$ where the unwritten entries are $0$s.