What is the rank of $A$?

Question

$A$ is a $15\times15$ matrix whose characteristic polynomial is $(x+1)^5(x-1)^3x^7$ and whose minimal polynomial is $(x+1)^3(x-1)^2x^3$. Then what will be the rank of $A$?

I think the rank of $A$ will be $8$ because rank is the minimum value.

Answer 1

The characteristic polynomial tells you about the eigenvalues and their multiplicity, but nothing that would let you guess the exact size of the eigenspaces.

The minimal polynomial now tells you the size of the biggest Jordan block for each eigenvalue, but still not the amount of blocks, which is what you want.

The nullity of $ T $ is the amount of Jordan blocks for the eigenvalue $0 $. The biggest has size $3\times 3 $, and the multiplicity is $7 $, so the extreme configurations are $3-3-1 $ (nullity $3 $) and $3-1-1-1-1 $ (nullity $5 $).

The nullity can be anything between $3 $ and $5 $.