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Let $A \in \mathbb{R}^{n \times n}$ be a symmetric matrix, such that $A$ is not of the form $A=c I_n, c \in \mathbb{R}$ and $(A-2I_n)^3 (A-3I_n)^4=0$. Find the minimal polynomial of $m_A(x)$of $A$.

I know that $m_A(x) | (x-2)^3(x-3)^4$, but I am stuck here. Any help appreciated.

mr.stealyourgirl
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2 Answers2

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Here is some Hints:

$1$. A Real Symmetric Matrix is always diagonalizable(Do you know that?)

$2$.A Matrix is diagonalizable $\Leftrightarrow$ its Minimal polynomial factors into distinct linear factor.

Hence $(x-2)(x-3)$ will be the Minimal polynomal.

Myshkin
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Hint

$1.$ The matrix is real and symmetric so this matrix is ....

$2.$ hence what we can say about the multiplicities $\alpha$ and $\beta$ of the root $2$ and $3$ of $m_A$?

$3$. Is it possible the case $\alpha=0$ or $\beta=0$?