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My friend was wondering, suppose the invertible matrices $A_i$, $1 \le i \le k$, are linearly independent as vectors in $\mathrm{M}_{n \times n}(\mathbb{R})$. Is it true that the $A_i^{-1}$ are linearly independent if $k \ge 3$?

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No. Take$$A_1=\begin{pmatrix}1&0&0\\0&\frac12&0\\0&0&\frac13\end{pmatrix},\ A_2=\begin{pmatrix}\frac12&0&0\\0&\frac13&0\\0&0&\frac14\end{pmatrix},\text{ and }A_3=\operatorname{Id}_3.$$They are linearly independent, but their inverses are not.

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