Relation between Algebraic multiplicities and rank of a matrix _A is a 6x6 matrix_ , $rank(A-3I) = 4$, the minimal polynomial of A is $(x-1)^2(x-3)^2$ I need to write the Jordan matrix options for A. How can I use the given information about the rank, what does it tell me about the eigenvalue 3 and his algebraic multiplicity in the characteristic polynomial? Can someone give me an elaborated explanation how it is connected? thank you

There are several things one needs to know to write down the jordan form. First, the algebraic multiplicity (degree of the root in the characteristic polynomial) tells you how many times that root apears on the diagonal. The geometric multiplicity is the dimension of the eigenspace inside the generalized eigenspace. The degree of the root in the minimal polynomial tells you the size of the largest jordan block.

Now the dimension of the nullspace of the $A-3I$ is the geometric multiplicity (dim eigenspace). Since the rank of that matrix is $4$, and it is a $6\times 6$ matrix, the nullspace must have dim $2$. So there are two jordan blocks corresponding to the eigenvalue $3$ and the largest one is $2\times 2$. Furthermore, the size of the largest jordan block for the eigenvalue $1$ is $2\times 2$. Can you write down the Jordan matrix?