Explain why the matrix \(Q= \left[\begin{array}{cccc} 1 & -2 & 8 & 5 \\ 1 & -1 & 4 & 3 \\ 1 & -1 & 5 & 4 \\ 1 & -1 & 0 & -1 \end{array}\right] \) is or is not invertible.
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} 1 & -2 & 8 & 5 \\ 1 & -1 & 4 & 3 \\ 1 & -1 & 5 & 4 \\ 1 & -1 & 0 & -1 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
\(Q\) is not invertible.Explain why the matrix \(Q= \left[\begin{array}{cccc} 1 & 4 & 5 & 4 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & 1 \\ 1 & 4 & 5 & 4 \end{array}\right] \) is or is not invertible.
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} 1 & 4 & 5 & 4 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & 1 \\ 1 & 4 & 5 & 4 \end{array}\right] = \left[\begin{array}{cccc} 1 & 4 & 0 & -1 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
\(Q\) is not invertible.Explain why the matrix \(P= \left[\begin{array}{cccc} -1 & 2 & -1 & 8 \\ -1 & 1 & -2 & 8 \\ -1 & 0 & -2 & 7 \\ 1 & 4 & 6 & -6 \end{array}\right] \) is or is not invertible.
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} -1 & 2 & -1 & 8 \\ -1 & 1 & -2 & 8 \\ -1 & 0 & -2 & 7 \\ 1 & 4 & 6 & -6 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \]
\(P\) is invertible.Explain why the matrix \(A= \left[\begin{array}{cccc} 4 & -2 & 6 & -2 \\ 3 & -5 & -6 & -5 \\ -1 & 0 & -3 & 0 \\ -3 & 5 & 6 & 5 \end{array}\right] \) is or is not invertible.
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} 4 & -2 & 6 & -2 \\ 3 & -5 & -6 & -5 \\ -1 & 0 & -3 & 0 \\ -3 & 5 & 6 & 5 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 3 & 0 \\ 0 & 1 & 3 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
\(A\) is not invertible.Explain why the matrix \(Q= \left[\begin{array}{cccc} 3 & 1 & 2 & -3 \\ -1 & 0 & -2 & 3 \\ 4 & 3 & -3 & 4 \\ 0 & 1 & -5 & 8 \end{array}\right] \) is or is not invertible.
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} 3 & 1 & 2 & -3 \\ -1 & 0 & -2 & 3 \\ 4 & 3 & -3 & 4 \\ 0 & 1 & -5 & 8 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & -2 \\ 0 & 0 & 1 & -2 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
\(Q\) is not invertible.Explain why the matrix \(M= \left[\begin{array}{cccc} 0 & 1 & -1 & -4 \\ -1 & 3 & -2 & -8 \\ 0 & 0 & 1 & 4 \\ 3 & -5 & 1 & 5 \end{array}\right] \) is or is not invertible.
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} 0 & 1 & -1 & -4 \\ -1 & 3 & -2 & -8 \\ 0 & 0 & 1 & 4 \\ 3 & -5 & 1 & 5 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \]
\(M\) is invertible.Explain why the matrix \(C= \left[\begin{array}{cccc} 2 & 2 & 7 & -3 \\ -2 & -1 & -4 & -3 \\ 1 & 0 & 1 & 3 \\ -1 & -1 & -3 & 1 \end{array}\right] \) is or is not invertible.
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} 2 & 2 & 7 & -3 \\ -2 & -1 & -4 & -3 \\ 1 & 0 & 1 & 3 \\ -1 & -1 & -3 & 1 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \]
\(C\) is invertible.Explain why the matrix \(Q= \left[\begin{array}{cccc} 1 & -5 & 4 & -6 \\ -1 & 6 & -5 & 7 \\ 2 & -6 & 4 & -8 \\ 1 & -1 & 0 & -2 \end{array}\right] \) is or is not invertible.
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} 1 & -5 & 4 & -6 \\ -1 & 6 & -5 & 7 \\ 2 & -6 & 4 & -8 \\ 1 & -1 & 0 & -2 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & -1 & -1 \\ 0 & 1 & -1 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
\(Q\) is not invertible.Explain why the matrix \(A= \left[\begin{array}{cccc} 0 & 0 & -2 & -6 \\ 0 & 0 & 1 & 3 \\ 1 & -4 & 2 & 6 \\ -2 & 8 & 0 & 0 \end{array}\right] \) is or is not invertible.
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} 0 & 0 & -2 & -6 \\ 0 & 0 & 1 & 3 \\ 1 & -4 & 2 & 6 \\ -2 & 8 & 0 & 0 \end{array}\right] = \left[\begin{array}{cccc} 1 & -4 & 0 & 0 \\ 0 & 0 & 1 & 3 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
\(A\) is not invertible.Explain why the matrix \(C= \left[\begin{array}{cccc} 1 & 1 & 0 & 1 \\ -3 & -2 & -2 & 5 \\ 4 & 4 & 1 & 1 \\ -1 & -1 & 2 & -6 \end{array}\right] \) is or is not invertible.
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} 1 & 1 & 0 & 1 \\ -3 & -2 & -2 & 5 \\ 4 & 4 & 1 & 1 \\ -1 & -1 & 2 & -6 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \]
\(C\) is invertible.Explain why the matrix \(C= \left[\begin{array}{cccc} -1 & 4 & 7 & -3 \\ 1 & 0 & 1 & 3 \\ 2 & -3 & -4 & 6 \\ 2 & -5 & -8 & 6 \end{array}\right] \) is or is not invertible.
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} -1 & 4 & 7 & -3 \\ 1 & 0 & 1 & 3 \\ 2 & -3 & -4 & 6 \\ 2 & -5 & -8 & 6 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 1 & 3 \\ 0 & 1 & 2 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
\(C\) is not invertible.Explain why the matrix \(B= \left[\begin{array}{cccc} -1 & 1 & 0 & 3 \\ -2 & -5 & 0 & -8 \\ -1 & -1 & 0 & -1 \\ 0 & 2 & 0 & 4 \end{array}\right] \) is or is not invertible.
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} -1 & 1 & 0 & 3 \\ -2 & -5 & 0 & -8 \\ -1 & -1 & 0 & -1 \\ 0 & 2 & 0 & 4 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & -1 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
\(B\) is not invertible.Explain why the matrix \(A= \left[\begin{array}{cccc} -1 & 1 & -3 & -4 \\ 0 & 1 & 3 & 3 \\ 2 & -4 & -1 & 1 \\ 3 & -6 & -7 & -4 \end{array}\right] \) is or is not invertible.
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} -1 & 1 & -3 & -4 \\ 0 & 1 & 3 & 3 \\ 2 & -4 & -1 & 1 \\ 3 & -6 & -7 & -4 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
\(A\) is not invertible.Explain why the matrix \(M= \left[\begin{array}{cccc} 1 & 1 & -3 & -4 \\ -2 & -1 & 5 & 5 \\ 2 & 0 & -4 & -2 \\ -1 & 0 & 2 & 1 \end{array}\right] \) is or is not invertible.
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} 1 & 1 & -3 & -4 \\ -2 & -1 & 5 & 5 \\ 2 & 0 & -4 & -2 \\ -1 & 0 & 2 & 1 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & -2 & -1 \\ 0 & 1 & -1 & -3 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
\(M\) is not invertible.Explain why the matrix \(M= \left[\begin{array}{cccc} 1 & -4 & -1 & -8 \\ 1 & -3 & 0 & -5 \\ 0 & 0 & 1 & 0 \\ -1 & 2 & -6 & 3 \end{array}\right] \) is or is not invertible.
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} 1 & -4 & -1 & -8 \\ 1 & -3 & 0 & -5 \\ 0 & 0 & 1 & 0 \\ -1 & 2 & -6 & 3 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \]
\(M\) is invertible.Explain why the matrix \(B= \left[\begin{array}{cccc} 1 & 1 & 1 & 3 \\ 2 & 3 & 1 & 2 \\ 1 & 4 & -1 & -4 \\ -3 & -4 & -2 & -4 \end{array}\right] \) is or is not invertible.
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} 1 & 1 & 1 & 3 \\ 2 & 3 & 1 & 2 \\ 1 & 4 & -1 & -4 \\ -3 & -4 & -2 & -4 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \]
\(B\) is invertible.Explain why the matrix \(A= \left[\begin{array}{cccc} 1 & -2 & 0 & -2 \\ -3 & 6 & 1 & 6 \\ -2 & 4 & -3 & 4 \\ 4 & -8 & 3 & -8 \end{array}\right] \) is or is not invertible.
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} 1 & -2 & 0 & -2 \\ -3 & 6 & 1 & 6 \\ -2 & 4 & -3 & 4 \\ 4 & -8 & 3 & -8 \end{array}\right] = \left[\begin{array}{cccc} 1 & -2 & 0 & -2 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
\(A\) is not invertible.Explain why the matrix \(P= \left[\begin{array}{cccc} -1 & 1 & 2 & -1 \\ -1 & 0 & 4 & -4 \\ 0 & 0 & 1 & -2 \\ -1 & 0 & 5 & -5 \end{array}\right] \) is or is not invertible.
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} -1 & 1 & 2 & -1 \\ -1 & 0 & 4 & -4 \\ 0 & 0 & 1 & -2 \\ -1 & 0 & 5 & -5 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \]
\(P\) is invertible.Explain why the matrix \(C= \left[\begin{array}{cccc} -1 & 0 & 3 & -4 \\ -2 & -3 & 1 & 5 \\ 1 & 1 & -1 & -1 \\ -2 & -4 & 0 & 8 \end{array}\right] \) is or is not invertible.
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} -1 & 0 & 3 & -4 \\ -2 & -3 & 1 & 5 \\ 1 & 1 & -1 & -1 \\ -2 & -4 & 0 & 8 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & -2 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & 1 & -2 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
\(C\) is not invertible.Explain why the matrix \(B= \left[\begin{array}{cccc} 1 & -1 & 3 & 1 \\ 1 & 0 & 5 & 7 \\ -1 & 0 & -4 & -5 \\ 2 & -2 & 2 & -6 \end{array}\right] \) is or is not invertible.
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} 1 & -1 & 3 & 1 \\ 1 & 0 & 5 & 7 \\ -1 & 0 & -4 & -5 \\ 2 & -2 & 2 & -6 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & -3 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
\(B\) is not invertible.Explain why the matrix \(Q= \left[\begin{array}{cccc} 4 & 2 & 2 & -2 \\ 1 & 3 & 8 & -3 \\ 1 & 0 & -1 & 0 \\ -2 & -2 & -4 & 2 \end{array}\right] \) is or is not invertible.
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} 4 & 2 & 2 & -2 \\ 1 & 3 & 8 & -3 \\ 1 & 0 & -1 & 0 \\ -2 & -2 & -4 & 2 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & -1 & 0 \\ 0 & 1 & 3 & -1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
\(Q\) is not invertible.Explain why the matrix \(P= \left[\begin{array}{cccc} 1 & 0 & 5 & 2 \\ 0 & 1 & -1 & 4 \\ -1 & 1 & -5 & 2 \\ 2 & -1 & 8 & 1 \end{array}\right] \) is or is not invertible.
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} 1 & 0 & 5 & 2 \\ 0 & 1 & -1 & 4 \\ -1 & 1 & -5 & 2 \\ 2 & -1 & 8 & 1 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \]
\(P\) is invertible.Explain why the matrix \(A= \left[\begin{array}{cccc} 0 & -1 & -4 & 5 \\ 1 & 0 & 0 & -1 \\ 1 & -1 & -3 & 3 \\ -2 & 1 & 3 & -2 \end{array}\right] \) is or is not invertible.
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} 0 & -1 & -4 & 5 \\ 1 & 0 & 0 & -1 \\ 1 & -1 & -3 & 3 \\ -2 & 1 & 3 & -2 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & -1 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
\(A\) is not invertible.Explain why the matrix \(P= \left[\begin{array}{cccc} 1 & -4 & -1 & 1 \\ -1 & 5 & 2 & -3 \\ 1 & -2 & 2 & -5 \\ 2 & -6 & 0 & -2 \end{array}\right] \) is or is not invertible.
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} 1 & -4 & -1 & 1 \\ -1 & 5 & 2 & -3 \\ 1 & -2 & 2 & -5 \\ 2 & -6 & 0 & -2 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & -1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & -2 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
\(P\) is not invertible.Explain why the matrix \(B= \left[\begin{array}{cccc} -5 & 3 & -7 & -3 \\ 3 & -2 & 5 & 3 \\ -3 & 2 & -4 & -2 \\ -2 & 2 & -4 & -4 \end{array}\right] \) is or is not invertible.
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} -5 & 3 & -7 & -3 \\ 3 & -2 & 5 & 3 \\ -3 & 2 & -4 & -2 \\ -2 & 2 & -4 & -4 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & -2 \\ 0 & 1 & 0 & -2 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
\(B\) is not invertible.Explain why the matrix \(Q= \left[\begin{array}{cccc} 1 & 0 & 1 & -2 \\ 5 & 1 & 0 & -6 \\ -3 & -1 & 3 & 1 \\ 4 & 1 & -3 & -2 \end{array}\right] \) is or is not invertible.
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} 1 & 0 & 1 & -2 \\ 5 & 1 & 0 & -6 \\ -3 & -1 & 3 & 1 \\ 4 & 1 & -3 & -2 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & -1 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
\(Q\) is not invertible.Explain why the matrix \(C= \left[\begin{array}{cccc} 1 & 1 & 0 & 3 \\ 1 & -2 & 6 & -3 \\ 1 & -2 & 6 & -3 \\ 0 & 4 & -8 & 8 \end{array}\right] \) is or is not invertible.
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} 1 & 1 & 0 & 3 \\ 1 & -2 & 6 & -3 \\ 1 & -2 & 6 & -3 \\ 0 & 4 & -8 & 8 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 2 & 1 \\ 0 & 1 & -2 & 2 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
\(C\) is not invertible.Explain why the matrix \(A= \left[\begin{array}{cccc} -1 & 2 & 0 & -1 \\ 0 & 1 & -1 & -2 \\ 0 & 2 & -2 & -4 \\ -2 & 0 & 4 & 6 \end{array}\right] \) is or is not invertible.
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} -1 & 2 & 0 & -1 \\ 0 & 1 & -1 & -2 \\ 0 & 2 & -2 & -4 \\ -2 & 0 & 4 & 6 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & -2 & -3 \\ 0 & 1 & -1 & -2 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
\(A\) is not invertible.Explain why the matrix \(N= \left[\begin{array}{cccc} 2 & -4 & -3 & 3 \\ -4 & 5 & 2 & -3 \\ 1 & -3 & -3 & 3 \\ -2 & 5 & 6 & -8 \end{array}\right] \) is or is not invertible.
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} 2 & -4 & -3 & 3 \\ -4 & 5 & 2 & -3 \\ 1 & -3 & -3 & 3 \\ -2 & 5 & 6 & -8 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \]
\(N\) is invertible.Explain why the matrix \(A= \left[\begin{array}{cccc} 0 & 2 & -1 & -6 \\ -2 & -3 & 4 & 7 \\ -2 & -3 & 5 & 4 \\ -1 & 0 & 1 & 0 \end{array}\right] \) is or is not invertible.
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} 0 & 2 & -1 & -6 \\ -2 & -3 & 4 & 7 \\ -2 & -3 & 5 & 4 \\ -1 & 0 & 1 & 0 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \]
\(A\) is invertible.Explain why the matrix \(B= \left[\begin{array}{cccc} -2 & 2 & -1 & -5 \\ -1 & 2 & 0 & 3 \\ 1 & -1 & 0 & 0 \\ -1 & 0 & 1 & 3 \end{array}\right] \) is or is not invertible.
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} -2 & 2 & -1 & -5 \\ -1 & 2 & 0 & 3 \\ 1 & -1 & 0 & 0 \\ -1 & 0 & 1 & 3 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \]
\(B\) is invertible.Explain why the matrix \(Q= \left[\begin{array}{cccc} 1 & 1 & -3 & -2 \\ 0 & 1 & -5 & -5 \\ 0 & 0 & 1 & 2 \\ 0 & 0 & 2 & 5 \end{array}\right] \) is or is not invertible.
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} 1 & 1 & -3 & -2 \\ 0 & 1 & -5 & -5 \\ 0 & 0 & 1 & 2 \\ 0 & 0 & 2 & 5 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \]
\(Q\) is invertible.Explain why the matrix \(C= \left[\begin{array}{cccc} 1 & 1 & -2 & -6 \\ 0 & 1 & 3 & 1 \\ 0 & -1 & -2 & 1 \\ 0 & 1 & 5 & 6 \end{array}\right] \) is or is not invertible.
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} 1 & 1 & -2 & -6 \\ 0 & 1 & 3 & 1 \\ 0 & -1 & -2 & 1 \\ 0 & 1 & 5 & 6 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \]
\(C\) is invertible.Explain why the matrix \(A= \left[\begin{array}{cccc} 1 & 2 & 3 & -6 \\ 0 & 1 & 0 & 1 \\ 1 & 7 & 4 & -3 \\ -1 & -2 & -2 & 4 \end{array}\right] \) is or is not invertible.
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} 1 & 2 & 3 & -6 \\ 0 & 1 & 0 & 1 \\ 1 & 7 & 4 & -3 \\ -1 & -2 & -2 & 4 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & -2 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & -2 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
\(A\) is not invertible.Explain why the matrix \(P= \left[\begin{array}{cccc} -2 & -1 & 3 & -5 \\ -1 & -1 & 3 & 0 \\ -1 & -1 & 4 & 1 \\ 0 & 2 & -2 & -5 \end{array}\right] \) is or is not invertible.
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} -2 & -1 & 3 & -5 \\ -1 & -1 & 3 & 0 \\ -1 & -1 & 4 & 1 \\ 0 & 2 & -2 & -5 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \]
\(P\) is invertible.Explain why the matrix \(B= \left[\begin{array}{cccc} 3 & 5 & -6 & 1 \\ -2 & -5 & 4 & 1 \\ 1 & 7 & -2 & -5 \\ 1 & 1 & -2 & 1 \end{array}\right] \) is or is not invertible.
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} 3 & 5 & -6 & 1 \\ -2 & -5 & 4 & 1 \\ 1 & 7 & -2 & -5 \\ 1 & 1 & -2 & 1 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & -2 & 2 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
\(B\) is not invertible.Explain why the matrix \(B= \left[\begin{array}{cccc} -1 & 4 & -6 & -2 \\ 0 & 1 & -2 & -1 \\ -2 & 6 & -8 & -2 \\ 1 & -4 & 6 & 2 \end{array}\right] \) is or is not invertible.
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} -1 & 4 & -6 & -2 \\ 0 & 1 & -2 & -1 \\ -2 & 6 & -8 & -2 \\ 1 & -4 & 6 & 2 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & -2 & -2 \\ 0 & 1 & -2 & -1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
\(B\) is not invertible.Explain why the matrix \(M= \left[\begin{array}{cccc} 1 & 4 & -3 & -3 \\ 2 & -3 & 0 & 8 \\ 0 & -2 & 1 & 3 \\ -1 & 3 & -1 & -4 \end{array}\right] \) is or is not invertible.
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} 1 & 4 & -3 & -3 \\ 2 & -3 & 0 & 8 \\ 0 & -2 & 1 & 3 \\ -1 & 3 & -1 & -4 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \]
\(M\) is invertible.Explain why the matrix \(A= \left[\begin{array}{cccc} -1 & 0 & 0 & -1 \\ 1 & -1 & -1 & -1 \\ -2 & 2 & 3 & 5 \\ 1 & -1 & -1 & -1 \end{array}\right] \) is or is not invertible.
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} -1 & 0 & 0 & -1 \\ 1 & -1 & -1 & -1 \\ -2 & 2 & 3 & 5 \\ 1 & -1 & -1 & -1 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & 1 & 3 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
\(A\) is not invertible.Explain why the matrix \(N= \left[\begin{array}{cccc} 1 & 1 & 0 & 2 \\ -2 & -1 & -2 & 1 \\ 2 & 2 & 1 & 0 \\ -2 & -2 & 1 & -7 \end{array}\right] \) is or is not invertible.
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} 1 & 1 & 0 & 2 \\ -2 & -1 & -2 & 1 \\ 2 & 2 & 1 & 0 \\ -2 & -2 & 1 & -7 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \]
\(N\) is invertible.Explain why the matrix \(A= \left[\begin{array}{cccc} 1 & 2 & 1 & -5 \\ 0 & 1 & 3 & -1 \\ 1 & 1 & -1 & -4 \\ 0 & 0 & 4 & 1 \end{array}\right] \) is or is not invertible.
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} 1 & 2 & 1 & -5 \\ 0 & 1 & 3 & -1 \\ 1 & 1 & -1 & -4 \\ 0 & 0 & 4 & 1 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \]
\(A\) is invertible.Explain why the matrix \(Q= \left[\begin{array}{cccc} 2 & 3 & 0 & -1 \\ -2 & 1 & -1 & 5 \\ 2 & -4 & 5 & -8 \\ -1 & -3 & 0 & -1 \end{array}\right] \) is or is not invertible.
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} 2 & 3 & 0 & -1 \\ -2 & 1 & -1 & 5 \\ 2 & -4 & 5 & -8 \\ -1 & -3 & 0 & -1 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & -2 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
\(Q\) is not invertible.Explain why the matrix \(A= \left[\begin{array}{cccc} 1 & 1 & -3 & 0 \\ -1 & 0 & 5 & 1 \\ 3 & 4 & -7 & 2 \\ -3 & -7 & 1 & 0 \end{array}\right] \) is or is not invertible.
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} 1 & 1 & -3 & 0 \\ -1 & 0 & 5 & 1 \\ 3 & 4 & -7 & 2 \\ -3 & -7 & 1 & 0 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & -5 & 0 \\ 0 & 1 & 2 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
\(A\) is not invertible.Explain why the matrix \(A= \left[\begin{array}{cccc} 1 & 2 & 3 & -5 \\ 0 & 1 & 2 & -2 \\ 0 & 4 & 8 & -8 \\ 0 & 4 & 8 & -8 \end{array}\right] \) is or is not invertible.
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} 1 & 2 & 3 & -5 \\ 0 & 1 & 2 & -2 \\ 0 & 4 & 8 & -8 \\ 0 & 4 & 8 & -8 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & -1 & -1 \\ 0 & 1 & 2 & -2 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
\(A\) is not invertible.Explain why the matrix \(P= \left[\begin{array}{cccc} -1 & -2 & 1 & -8 \\ -2 & -3 & 7 & 5 \\ -1 & -1 & 3 & 3 \\ -1 & -2 & 3 & -1 \end{array}\right] \) is or is not invertible.
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} -1 & -2 & 1 & -8 \\ -2 & -3 & 7 & 5 \\ -1 & -1 & 3 & 3 \\ -1 & -2 & 3 & -1 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \]
\(P\) is invertible.Explain why the matrix \(B= \left[\begin{array}{cccc} 1 & 2 & 2 & 5 \\ 0 & 1 & 2 & 5 \\ 0 & -1 & -1 & -2 \\ 2 & 2 & 2 & 6 \end{array}\right] \) is or is not invertible.
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} 1 & 2 & 2 & 5 \\ 0 & 1 & 2 & 5 \\ 0 & -1 & -1 & -2 \\ 2 & 2 & 2 & 6 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & 1 & 3 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
\(B\) is not invertible.Explain why the matrix \(Q= \left[\begin{array}{cccc} 0 & 1 & 7 & 8 \\ 0 & 1 & 2 & 3 \\ 0 & 0 & 1 & 1 \\ -1 & -1 & 1 & 0 \end{array}\right] \) is or is not invertible.
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} 0 & 1 & 7 & 8 \\ 0 & 1 & 2 & 3 \\ 0 & 0 & 1 & 1 \\ -1 & -1 & 1 & 0 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
\(Q\) is not invertible.Explain why the matrix \(C= \left[\begin{array}{cccc} 0 & 1 & -1 & 3 \\ -1 & -1 & -1 & -3 \\ 2 & 3 & 2 & 8 \\ 1 & 1 & 4 & 0 \end{array}\right] \) is or is not invertible.
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} 0 & 1 & -1 & 3 \\ -1 & -1 & -1 & -3 \\ 2 & 3 & 2 & 8 \\ 1 & 1 & 4 & 0 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 2 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 0 \end{array}\right] \]
\(C\) is not invertible.Explain why the matrix \(M= \left[\begin{array}{cccc} 4 & 0 & 7 & 4 \\ -2 & 1 & -4 & -3 \\ -1 & 0 & 0 & 5 \\ 1 & 0 & 2 & 2 \end{array}\right] \) is or is not invertible.
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} 4 & 0 & 7 & 4 \\ -2 & 1 & -4 & -3 \\ -1 & 0 & 0 & 5 \\ 1 & 0 & 2 & 2 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \]
\(M\) is invertible.Explain why the matrix \(P= \left[\begin{array}{cccc} -2 & 3 & 3 & 1 \\ 3 & -5 & -6 & -2 \\ 2 & -2 & 1 & 2 \\ 3 & -4 & -3 & 0 \end{array}\right] \) is or is not invertible.
Answer:
\[\operatorname{RREF} \left[\begin{array}{cccc} -2 & 3 & 3 & 1 \\ 3 & -5 & -6 & -2 \\ 2 & -2 & 1 & 2 \\ 3 & -4 & -3 & 0 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \]
\(P\) is invertible.