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