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Let $m,n\in \mathbb{N},\ m>n\ge 1$ and the matrices $A_1,A_2,...,A_m\in \mathcal{M}_n(\mathbb{C})$ such that the matrix $\sum_{i=1}^mA_i$ is nonsingular. Prove that there is a set $S\subset \{1,2,...,m\}$ with at most $n$ elements, such that $\sum_{i\in S}A_i$ is nonsingular. Any help,please?

Mohammed M. Zerrak
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BlueSyrup
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  • Hint: A matrix is non-singular only if its columns are linearly independent. What are the columns of $\sum_{i=1}^mA_i$? – TheEmptyFunction Mar 10 '20 at 17:17
  • Can you elaborate? I see that there are a lot of linearly independent columns, but i do not understand why that implies that $ \sum_{j=1}^mC_{ij} $ with $ i={1,2..n} $ are linearly dependent. – BlueSyrup Mar 10 '20 at 21:14
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    Try thinking about the following: Let $A=\sum_{i=1}^mA_i$, let $C^i_j$ be the $j^{th}$ column of $A_i$, and let $C_j$ be the $j^{th}$ column of $A$. Since A is non-singular, then we can write any vector in $\mathbb{C}^n$ as a linear combination of its columns. In particular, we can write $C_1^i=a^i_1C_1+\cdots+a^i_nC_n$. But we also know that $\sum_{i=1}^mC_^i_1=C_1$. This additional equation and the fact that $C_1=\alpha_1 C_1+\cdots +\alpha_n C_n\Leftrightarrow \alpha_j=\begin{cases}1&j=1\0&\text{ else}\end{cases}$ will force a constraint on the coefficients $a^i_j$. – TheEmptyFunction Mar 10 '20 at 23:34

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