Explain why the matrix \(P= \left[\begin{array}{cccc} -3 & 2 & -6 & 4 \\ -2 & 1 & -3 & 3 \\ 5 & 0 & 1 & -8 \\ 0 & 3 & -6 & 4 \end{array}\right] \) is or is not invertible.
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} -3 & 2 & -6 & 4 \\ -2 & 1 & -3 & 3 \\ 5 & 0 & 1 & -8 \\ 0 & 3 & -6 & 4 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \]
\(P\) is invertible.