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I got this question from somewhere. It is easy but I am not able to find the incorrect option out of these.

Let $A$ be a square matrix of order $n$. Let $m_A$ denote the minimal polynomial and $c_A$ denote the characteristic polynomial then, Find the incorrect option(s)

  1. $\lambda$ is an eigenvalue $\iff m_{A}(\lambda)=0.$

  2. The minimal polynomial of a matrix is unique

  3. If $P(x)$ is a polynomial such that $P(A)=0\implies m_{A}|p_A$

  4. $m_A(x)|c_A(x)$

  5. Every irreducible factor of $c_A(x)$ is also a factor of $m_A(x)$ and conversely

  6. Every root of $c_A(x)$ is also a root of $m_A(x)$ and conversely

  7. $\deg(m_A(x))\geq $ number of distinct eigenvalues.

My Efforts

$(1).(2),(3), (4),(6)$ These are all true as I have proved them.

I have doubts in $(5)$ and $(7)$

I need some hints. I am stuck.

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Shweta Aggrawal
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1 Answers1

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It looks like they're all true. For $(5)$ look here. $(7)$ looks easy.