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