3.1 Matrix operations

Matrix multiplication

• multiplying a matrix by another
  • you cannot multiply matrices in different dimensions

• matrix has the same rules as multiplication however you cant

  • AB=/=BA
  • AB=AC, B=/=c
  • AB=0, it doesn’t mean that either A or B is zero

• how to multiply matrices

$$\left[\begin{array}{ccc}a& b& c\\ x& y& z\end{array}\right]\ast \left[\begin{array}{cc}l& m\\ n& o\\ p& q\end{array}\right]⟹\left[\begin{array}{cc}al+bn+cp& am+bo+cq\\ xl+yn+zp& xm+yo+zq\end{array}\right]$$

Ax=b

• Example 1

$$\begin{array}{r}x+y+3z=5\\ x-z=7\end{array}$$ $$⟹\left[\begin{array}{ccc}1& 1& 3\\ 1& 0& -1\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}5\\ 7\end{array}\right]$$

• This is the Ax=b form (EASY)