Set of Matrices and Invertibility If $S= \{A_1,\cdots ,A_k\}$ is a set of row equivalent matrices, and there's a linear combination of the elements of $S$ which makes an invertible matrix, then I need to show that every one of the elements of $S$ is invertible.
I've been struggling with the way to tackle this problem (tried induction, by contradiction, etc..) but achieved very little results. Would appreciate your help.

Hint:
Let A be an invertible matrix which is a linear combination of the elements of S.
If $A_{i}x=0$ for some i, then $A_{j}x=0$ for $1\le j \le k$ since the elements of S are row-equivalent; so $Ax=0$.
