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Matrix Theory: Linear Algebra and Beyonds

Peter Tan

Section 1.3 General solutions

Example 1.3.1.

Find the general solution of a linear system
\begin{align*} 3x_2-6x_3+6x_4+4x_5\amp=-5\\ 3x_1-7x_2+8x_3-5x_4+8x_5\amp=9\\ 3x_1-9x_2+12x_3-9x_4+6x_5\amp=15 \end{align*}
  1. The matrix \(C\) can be interpreted as
    \begin{align*} x_1-2x_3+3x_4\amp=-24\\ x_2-2x_3+2x_4\amp=-7\\ x_5\amp=4 \end{align*}
  2. The general solution can be written as
    \begin{align*} x_1\amp=-24+2x_3-3x_4\\ x_2\amp=-7+2x_3-2x_4\\ x_5\amp=4 \end{align*}
  3. The general solution is:
    \begin{equation*} \left[\begin{array}{r} x_1 \\ x_2\\ x_3\\ x_4\\ x_5 \end{array}\right]=\left[\begin{array}{c} -24+2x_3-3x_4\\ -7+2x_3-2x_4\\ x_3\\ x_4\\ 4 \end{array}\right]=\left[\begin{array}{c} -24\\ -7\\ 0\\ 0\\ 4 \end{array}\right]+x_3\left[\begin{array}{c} 2\\ 2\\ 1\\ 0\\ 0 \end{array}\right]+x_4\left[\begin{array}{c} -3\\ -2\\ 0\\ 1\\ 0 \end{array}\right] \end{equation*}
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