Explain why the matrix \(B= \left[\begin{array}{cccc} 3 & 2 & 5 & 2 \\ 2 & 1 & 1 & -3 \\ 1 & 1 & 3 & 3 \\ 0 & 0 & -3 & -6 \end{array}\right] \) is or is not invertible.
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} 3 & 2 & 5 & 2 \\ 2 & 1 & 1 & -3 \\ 1 & 1 & 3 & 3 \\ 0 & 0 & -3 & -6 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & -2 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
\(B\) is not invertible.