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 Post subject: DeterminantPosted: Thu Jun 09, 2016 12:04 pm

Joined: Sat Nov 07, 2015 6:12 pm
Posts: 828
Location: Larisa
Let $A\in \mathcal{M}_n\left ( \mathbb{C} \right )$ with $n\geq 2$. If $\det \left ( A+X \right )=\det A+\det X$ for every matrix $X \in \mathcal{M}_n\left ( \mathbb{C} \right )$ , then prove that $A=\mathbb{O}$.

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 Post subject: Re: DeterminantPosted: Thu Jun 09, 2016 12:05 pm

Joined: Mon Nov 09, 2015 11:52 am
Posts: 76
Location: Limassol/Pyla Cyprus
Suppose that $A \neq 0$, say $A_{ij} \neq 0$ for some $i,j$. Let $P$ be any permutation matrix with $P_{ij}=1$ and let $Q$ be the matrix obtained from $P$ by changing its $ij$-entry to $0$. Finally let $X = xQ$ where $x \in \mathbb{C}$.

We have that $\det(X) = 0$ and that $\det(X) = \det(A+X) - \det(A)$ is a polynomial in $x$. Furthermore, the coefficient of $x^{n-1}$ of this polynomial is $\pm A_{ij}$ depending on the sign of the corresponding permutation. So the polynomial is not identically zero, a contradiction.

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