Suppose that I have a $n\times t$ matrix $\boldsymbol{X}$ that is full rank and a non-singular matrix $\boldsymbol{L} = \begin{bmatrix} \boldsymbol{L}_1 & \boldsymbol{L}_2 \end{bmatrix}$ such that $\boldsymbol{L}_1$ and $\boldsymbol{L}_2$ are $n\times t$ and $n\times (n-t)$ matrices, respectively, such that $\boldsymbol{L}_2^\top\boldsymbol{L}_2 = \boldsymbol{I}_{n-t}$, $\boldsymbol{L}_1^\top\boldsymbol{L}_2 = \boldsymbol{0}$, $\boldsymbol{L}_1^\top\boldsymbol{X} = \boldsymbol{I}_t$ and $\boldsymbol{L}_2^\top\boldsymbol{X} = \boldsymbol{0}$.
How do you prove that $\begin{bmatrix} \boldsymbol{X} & \boldsymbol{L}_2 \end{bmatrix}$ is non-singular?
Compute:
$$\begin{bmatrix} L_1^T \\ L_2^T \end{bmatrix} \begin{bmatrix} X & L_2 \end{bmatrix}$$