Skip to main content

Chapter 2 numbering bug?

the following is an <ol> but numbers don't appear

  1. \begin{equation*} a^2+b^2=c^2 \end{equation*}

  2. \begin{equation*} a^2+b^2=c^2 \end{equation*}

  1. \begin{equation*} \begin{matrix} x & + & 2 \, y & + & z & = & 2 \\ 3 \, x & + & 4 \, y & & & = & 9 \\ & - & y & - & 2 \, z & = & 3 \\ -3 \, x & - & 4 \, y & - & z & = & -6 \\ \end{matrix} \end{equation*}

  2. \begin{equation*} \left[\begin{array}{ccc|c} 1 & 2 & 1 & 2 \\ 3 & 4 & 0 & 9 \\ 0 & -1 & -2 & 3 \\ -3 & -4 & -1 & -6 \end{array}\right] \end{equation*}

Consider the vector equation.

\begin{equation*} x_{1} \left[\begin{array}{c} 1 \\ 3 \\ 0 \\ -3 \end{array}\right] + x_{2} \left[\begin{array}{c} 2 \\ 4 \\ -1 \\ -4 \end{array}\right] + x_{3} \left[\begin{array}{c} 1 \\ 0 \\ -2 \\ -1 \end{array}\right] = \left[\begin{array}{c} 2 \\ 9 \\ 3 \\ -6 \end{array}\right] \end{equation*}

  1. Write a system of scalar equations corresponding to this vector equation.

  2. Write an augmented matrix corresponding to this vector equation.

Solution.

  1. \begin{equation*} \begin{matrix} x & + & 2 \, y & + & z & = & 2 \\ 3 \, x & + & 4 \, y & & & = & 9 \\ & - & y & - & 2 \, z & = & 3 \\ -3 \, x & - & 4 \, y & - & z & = & -6 \\ \end{matrix} \end{equation*}

  2. \begin{equation*} \left[\begin{array}{ccc|c} 1 & 2 & 1 & 2 \\ 3 & 4 & 0 & 9 \\ 0 & -1 & -2 & 3 \\ -3 & -4 & -1 & -6 \end{array}\right] \end{equation*}

Consider the scalar system of equations

\begin{alignat*}{5} 3x_1 &\,+\,& 2x_2 &\,\,& &\,+\,&x_4 &= 1 \\ -x_1 &\,-\,& 4x_2 &\,+\,&x_3&\,-\,&7x_4 &= 0 \\ &\,\,& x_2 &\,-\,&x_3 &\,\,& &= -2 \end{alignat*}

  1. Rewrite this system as a vector equation.

  2. Write an augmented matrix corresponding to this system.

Solution.

  1. \begin{equation*} x_1\left[\begin{array}{c} 3 \\ -1 \\ 0 \end{array}{}\right] + x_2 \left[\begin{array}{c}2 \\ -4 \\ 1 \end{array}{}\right]+ x_3 \left[\begin{array}{c} 1 \\ 1 \\ -1 \end{array}{}\right] + x_4 \left[\begin{array}{c} 1 \\ -7 \\ 0 \end{array}{}\right] = \left[\begin{array}{c} 1 \\ 0 \\ -2 \end{array}{}\right] \end{equation*}

  2. \begin{equation*} \left[\begin{array}{cccc|c} 3 & 2 & 0 & 1 & 1 \\ -1 & -4 & 1 & -7 & 0 \\ 0 & 1 & -1 & 0 & -2 \end{array}\right] \end{equation*}