Explain why the matrix \(C= \left[\begin{array}{cccc} 1 & 2 & 1 & -3 \\ -4 & -7 & -3 & 7 \\ -2 & -4 & -1 & 2 \\ -1 & -1 & 1 & -5 \end{array}\right] \) is or is not invertible.
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} 1 & 2 & 1 & -3 \\ -4 & -7 & -3 & 7 \\ -2 & -4 & -1 & 2 \\ -1 & -1 & 1 & -5 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \]
\(C\) is invertible.