For the matrix A where

$$

A = \begin{bmatrix} 1 & 2 & -1 & 1 \\ 2 & 1 & 0 & 3 \\ 0 & -3 & 2 & 1 \\ -3 & 0 & -1 & -5 \end{bmatrix}

$$

find matrices P and Q such that

$$

PAQ = \begin{bmatrix} I_r & 0 \\ 0 & 0 \end{bmatrix},

$$

where r is the rank of A. This demonstrates Theorem 1.6.8.