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Linear Algebra: Example Sheet 2 of 4 - DPMMS
14. Let $A, B$ in $\mathcal{M}_n(\mathbb{R})$ such that $\exists X \in \operatorname{Ker} A \backslash\{0\}$ with $B X \in \operatorname{Im} A$. Let $A_i$ be the matrix obtained by replacing the $i$-th column of $A$ by the $i$-th column of $B$. Show that $\sum_{i=1}^n \operatorname{det} A_i=0$. |
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