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Matrix Theory: Linear Algebra and Beyonds

Peter Tan

Section 1.5 Exercise

For each of the following linear systems, determine whether the system is consistent, and, if so, find the general solutions.
  1. \begin{equation*} \begin{alignedat}{5} \amp{}{}\amp 2x_2\amp {}+{}\amp 2x_3 {}={} \amp -4 \\ 2x_1\amp{}-{}\amp x_2\amp {}+{}\amp 3x_3 {}={} \amp 10 \\ -x_1\amp{}+{}\amp x_2\amp\amp {}={} \amp 8 \\ 3x_1\amp{}+{}\amp 2x_2\amp {}-{}\amp x_3 {}={} \amp -118 \\ \end{alignedat} \end{equation*}
  2. \begin{equation*} \begin{alignedat}{5} -x_1\amp{}+{}\amp x_2\amp \amp \amp {}+{} \amp 2x_4 \amp {}={} \amp -1 \\ 2x_1\amp{}-{}\amp x_2\amp {}+{}\amp 3x_3 \amp {}+{} \amp 3x_4 \amp {}={} \amp 8 \\ \amp{}{}\amp 2x_2\amp {}+{}\amp 2x_3 \amp {}+{} \amp 6x_4 \amp {}={} \amp 4 \\ 3x_1\amp{}+{}\amp 2x_2\amp {}-{}\amp x_3 \amp {}-{} \amp 3x_4 \amp {}={} \amp 1 \\ \end{alignedat} \end{equation*}
  3. \begin{equation*} \begin{alignedat}{5} 2x_1\amp{}-{}\amp x_2\amp {}+{}\amp 3x_3 \amp {}+{} \amp 3x_4 \amp {}={} \amp 8 \\ 3x_1\amp{}+{}\amp 2x_2\amp {}-{}\amp x_3 \amp {}-{} \amp 3x_4 \amp {}={} \amp 0 \\ -x_1\amp{}+{}\amp x_2\amp \amp \amp {}+{} \amp 2x_4 \amp {}={} \amp -1 \\ \amp{}{}\amp 2x_2\amp {}+{}\amp 2x_3 \amp {}+{} \amp 6x_4 \amp {}={} \amp 4 \\ \end{alignedat} \end{equation*}
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