# Rank of a linear operator given its characteristic and minimal polynomial [closed]

Let $T\colon V \to V$ be a linear operator on the vector space $V$ with characteristic polynomial $\lambda^4(\lambda-4)^5$ and minimal polynomial $\lambda(\lambda-4)$, then what is the rank of $T$?

## closed as off-topic by Xander Henderson, Don Thousand, ArsenBerk, Toby Mak, I am BackOct 20 '18 at 8:20

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• So, what is your 'problem'? – GAVD Dec 3 '15 at 8:04
• Have you tried anything? – ASKASK Dec 3 '15 at 8:07
• Yes but not worked. – sharafat salam Dec 3 '15 at 8:16
• Use Jordon form – Black-horse Dec 3 '15 at 9:14
• Can you do it black horse? – sharafat salam Dec 3 '15 at 9:26

The Jordon form of that matrix will be $$\begin{bmatrix} (0)_4&0\\ 0&(4)_5\\ \end{bmatrix}$$ where $(0)_4$ is $4\times 4$ zero matrix and $(4)_5$ is $5\times 5$ diagonal matrix with diagonal entries $4$. Hence the rank is $5$
The minimal polynomial is of the form $$(x-a)(x-b)$$ hence the matrix is diagonalizable and hence rank is 5