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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 \( A=\mathbb{O} \).

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.