# Questions tagged [characteristic-polynomial]

The characteristic polynomial of a square matrix is a polynomial that is invariant under matrix similarity and has the eigenvalues as roots.

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### Is the Jordan normal form uniquely determined by the characteristic and minimal polynomial if they are the same?

I'm studying for an exam and I can't get anywhere with a problem. I've seen similar questions on here but not the same. The problem provides the characteristic polynomial $XA(x) = (x-3)^2(x-1)$ and ...
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### Classification of all similar $3\times 3$ matrices over $\mathbb{R}$

I want to ask a question regarding classes of similar $3 \times 3$ matrices. We were told that two matrices are similar if and only if they have the same Jordan normal form. That led me to trying to ...
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### Error in the book Graph Theory: Calculate this determinant/characteristic polynomial

In the book "Graph Theory" - by Bondy and Murty exercise 1.1.22 b) ii) wants me to prove that $$det(J-(1+\lambda)I_n) = (1+\lambda-n)(1+\lambda)^{n-1}$$ where $J$ is the $n$ x $n$ matrix ...
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### Let $A \in M_n(\mathbb R)$ such that $A^2+A+5I_n = 0$. Find the characteristic polynomial of $A$ [duplicate]

Let $A \in M_n(\mathbb R)$ such that $$A^2+A+5I_n = 0$$ Find the characteristic polynomial of $A$. I tried two different approaches and got stuck on both. I am wondering if I was even headed in the ...
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### Proof of Cayley-Hamilton theorem over any field $\Bbb K$

I'm currently studying the Cayley-Hamilton theorem for an exam, and I do not quite get the proof presented in the lecture. It was structured as follows: first we'll prove it over $\mathbb{C}$ using ...
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### If two matrices have the same characteristic polynomial then do they have the same determinant?

There is a similar question here, but it's asking that if two matrices have the same characteristic polynomial then are they similar. If the answer were positive then the answer to my question will ...
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### Let $a$ be a sequence where $a_0 = 21$, $a_1 = 35$, and $a_{n+2} = 4a_{n+1} − 4a_n + n^2$ for $n ≥ 2$. Compute $a_{2006} \pmod {100}$

Problem Let $a_0, a_1, a_2, ...$ be a sequence of real numbers defined by $a_0 = 21, a_1 = 35$, and $a_{n+2} = 4a_{n+1}-4a_n+n^2$ for $n ≥ 2$. Compute the remainder obtained when $a_{2006}$ is ...
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### Dimension of image(T) and characteristic polynomial

Let $V$ be a finite dimensional vector space and let $T: V \to V$ be a diagonalizable linear map with characteristic polynomial $$f ( x ) = x^2(x-3)^2(x-9)(x+2)$$ Find the dimension of the image of T. ...
### An $n\times n$ matrix, $n\ge 2$ with characteristic polynomial $x^{n-2}(x^2-1)$ [duplicate]
$A$ is an $n\times n$ matrix, $n\ge 2$ with characteristic polynomial $x^{n-2}(x^2-1)$. Then, which of the following is true? $A^n=A^{n-2}$ rank of $A$ is $2$ rank of $A$ is atleast $2$ there are ...