Explain why the matrix \(N= \left[\begin{array}{cccc} 1 & -1 & -7 & -4 \\ 1 & 0 & -3 & -1 \\ -1 & -1 & 0 & -2 \\ 0 & 2 & 4 & 6 \end{array}\right] \) is or is not invertible.
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} 1 & -1 & -7 & -4 \\ 1 & 0 & -3 & -1 \\ -1 & -1 & 0 & -2 \\ 0 & 2 & 4 & 6 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & -1 \\ 0 & 1 & 0 & 3 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
\(N\) is not invertible.