Question

Let \(A=\left(\begin{array}{rrr}1 & 1 & -1 \\ 1 & -1 & 2\end{array}\right)\)

a) Find the general solution \(\mathbf{Z}\) of the homogeneous equation \(A \mathbf{Z}=0\).
b) Find some solution of \(A \mathbf{X}=\left(\begin{array}{l}1 \\ 2\end{array}\right)\)
c) Find the general solution of the equation in part b).
d) Find some solution of \(A \mathbf{X}=\left(\begin{array}{l}-1 \\ -2\end{array}\right)\) and of \(A \mathbf{X}=\left(\begin{array}{l}3 \\ 6\end{array}\right)\)
e) Find some solution of \(A \mathbf{X}=\left(\begin{array}{l}3 \\ 0\end{array}\right)\)
f) Find some solution of \(A \mathbf{X}=\left(\begin{array}{l}7 \\ 2\end{array}\right)\). [Note: \(\left(\begin{array}{l}7 \\ 2\end{array}\right)=\left(\begin{array}{l}1 \\ 2\end{array}\right)+2\left(\begin{array}{l}3 \\ 0\end{array}\right)\) ].
[Remark: After you have done parts a), b) and e), it is possible immediately to write the solutions to the remaining parts.]