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  • MAT223 Readings 01

Row Operations in Linear Algebra

• You have three elementary operations to a System of Linear Equations.
  • 1: Exchange the order of any two equations
  • 2: Multiply both sides of an equation with a non-zero constant
  • 3: Add some multiple of one equation onto another equation.
• Example:
  • Solve the System of Linear Equations using elementary row operations.
  • $$\begin{array}{cc}x+y& =2\\ x-y& =4\end{array}$$
  • (2) + (1)
  • $$\begin{array}{cc}x+y& =2\\ x-y+x+y& =4+2\end{array}$$
  • $$\begin{array}{cc}x+y& =2\\ 2x& =6\end{array}$$
  • (2)/2
  • $$\begin{array}{cc}x+y& =2\\ x& =3\end{array}$$
  • (1) - (2)
  • $$\begin{array}{cc}x+y-x& =2-3\\ x& =3\end{array}$$
  • $$\begin{array}{ll}y& =-1\\ x& =3\end{array}$$
• Ex:
  • Do the following row operations on the matrix:
  • $\left[\begin{array}{cccc}1& -2& 1& 1\\ 1& 6& -4& 4\end{array}\right]$
  • Swap row 1 and 2
  • $\left[\begin{array}{cccc}1& 6& -4& 4\\ 1& -2& 1& 1\end{array}\right]$
  • Multiply row 2 by 2
  • $\left[\begin{array}{cccc}1& 6& -4& 4\\ 2& -4& 2& 2\end{array}\right]$
  • Add 2 copies of row 1 to row 2
  • $\left[\begin{array}{cccc}1& 6& -4& 4\\ 4& 8& -6& 10\end{array}\right]$