1.4.30 Matrix transformations

Condition: Let \( A_{1}, A_{2}, \ldots A_{n+1}- \) be matrices of size \( n \times n \). Prove that there are \( n+1 \) numbers \( x_{1}, x_{2}, \ldots, x_{n+1} \) not simultaneously equal to zero such that the matrix \( x_{1} A_{1}+\cdots+x_{n+1} A_{n+1} \) degenerate.