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Let $F$ be a field and let $V = F^{4\times4}$ be the vector space of $4x4$ matrices over $F$. For $A \in F^{4\times4}$, define $T_A : V \rightarrow V$ by $T_A(B) = AB$ for each $B \in V$.

Question: True or false "The minimal polynomial of $T_A$ is never equal to the characteristic polynomial of $T_A$"

I know how to prove that the minimal polynomial of $T_A$ is equal to the minimal polynomial of $A$. I was thinking I could pick a suitable diagonal matrix $A$ to make the question false but after testing a couple of examples I am failing to come up with the correct diagonal matrix.

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I know how to prove that the minimal polynomial of $T_A$ is equal to the minimal polynomial of $A$.

Good, then you are almost done. The minimal polynomial of $A$ has degree at most [blank], while the characteristic polynomial of $T_A$ has degree equal to [blank]. You can fill in the blanks by looking at the dimensions of the spaces which $A$ and $T_A$ act on.

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Thanks for your help. I do not understand your last statement about considering the dimension of the spaces that $A$ and $T_A$ act on could you please clarify. –  user7980 Aug 8 '11 at 7:50
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@user7980: The degree of the characteristic polynomial is the dimension of the space being acted on, and the degree of the minimal polynomial is at most the degree of the characteristic polynomial. The two spaces involved have different dimensions, so if the minimal polynomials are the same, they can't be equal to the characteristic polynomial of the higher degree. –  joriki Aug 8 '11 at 7:58
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@user7980: If you have an $n$-by-$n$ matrix, then you can think of it as a linear transformation on $F^n$. You know the degree of the characteristic polynomial of an $n$-by-$n$ matrix? By the Cayley-Hamilton theorem, the minimal polynomial has equal or lesser degree. You can apply this to $A$. Now $T_A$ is not explicitly defined as a matrix, but as a linear tranformation on a given vector space $F^{4\times 4}$. What's the degree of the char. polynomial of a linear transformation acting on a space of this size? (It may help to think of the size of the corresponding matrix of $T_A$) –  Jonas Meyer Aug 8 '11 at 8:01
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