Let $ A=\left(a_{i j}\right)_{i, j=1, \ldots, n} \in \mathbb{R}^{n \times n} $ be an invertible matrix.
Show that an $ L R $-decomposition of $ A $ exists if and only if $ \operatorname{det}\left(A^{[k]}\right) \neq 0 $ for all $k=1, \ldots, n $, where $ A^{[k]} \in \mathbb{R}^{k \times k}, 1 \leq k \leq n $ is defined as $ A^{[k]}:=\left(\begin{array}{ccc} a_{11} & \cdots & a_{1 k} \\ \vdots & \ddots & \vdots \\ a_{k 1} & \cdots & a_{k k} \end{array}\right) \in \mathbb{R}^{k \times k} $
Hint: You may use the following theorem without proof:
Let $ A \in \mathbb{R}^{n \times n} $ be given. Then for each lower triangular matrix $ L \in \mathbb{R}^{n \times n} $ it holds $ (L \cdot A)^{[k]}=L^{[k]} \cdot A^{[k]}, \quad k=1, \ldots, n $.