Explain why the matrix \(M= \left[\begin{array}{cccc} 1 & -1 & -2 & 2 \\ -2 & 3 & 3 & -5 \\ 2 & -5 & -1 & 7 \\ 2 & -5 & -1 & 7 \end{array}\right] \) is or is not invertible.
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} 1 & -1 & -2 & 2 \\ -2 & 3 & 3 & -5 \\ 2 & -5 & -1 & 7 \\ 2 & -5 & -1 & 7 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & -3 & 1 \\ 0 & 1 & -1 & -1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
\(M\) is not invertible.