Explain why the matrix \(P= \left[\begin{array}{cccc} 1 & 4 & -6 & -4 \\ 0 & 1 & -1 & -2 \\ 0 & 2 & -1 & -6 \\ 0 & 4 & -7 & -1 \end{array}\right] \) is or is not invertible.
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} 1 & 4 & -6 & -4 \\ 0 & 1 & -1 & -2 \\ 0 & 2 & -1 & -6 \\ 0 & 4 & -7 & -1 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \]
\(P\) is invertible.