1Matrices

Matrix: a rectangular array of numbers, arranged in rows and columns, like a spreadsheet, e.g.

This matrix has two rows and three columns.

• row matrix: $\left(\begin{array}{ccc}1& 2& 3\end{array}\right)$
• column matrix:$\left(\begin{array}{c}1\\ 2\end{array}\right)$
• square matrix: $\left(\begin{array}{cc}1& 2\\ 3& 4\end{array}\right)$
• zero matrix: $\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\end{array}\right)$
• corresponding entries: entries in different matrices but the same row and column, e.g. $\left(\begin{array}{ccc}\hfill 6& 2& -1\\ -2& 0& 5\end{array}\right)\left(\begin{array}{ccc}11& 12& 13\\ 21& 22& 23\end{array}\right)$
• Matrices are equal if their corresponding entries are equal: $\left(\begin{array}{cc}2+3& 4\end{array}\right)=\left(\begin{array}{cc}5& 2^2\end{array}\right)$

• Adding or subtracting matrices

  • Matrices must have same dimensions
  • Just add or subtract corresponding entries
  • $\left(\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\end{array}\right)+\left(\begin{array}{ccc}2& 3& 4\\ 5& 6& 7\end{array}\right)=\left(\begin{array}{ccc}1+2& 2+3& 3+4\\ 4+5& 5+6& 6+7\end{array}\right)=\left(\begin{array}{ccc}3& 5& 7\\ 9& 11& 13\end{array}\right)$
• A scalar is just a number (as distinguished from a matrix)
• Scalar multiplication: multiply each entry by the scalar:
• $3\cdot \left(\begin{array}{cc}0& 1\\ -2& 4\end{array}\right)=\left(\begin{array}{cc}3\cdot 0& 3\cdot 1\\ 3\cdot (-2)& 3\cdot 4\end{array}\right)=\left(\begin{array}{cc}0& 3\\ -6& 12\end{array}\right)$

• Algebraic properties follow from similar properties of scalars:

  • Associative property of addition: $(A+B)+C=A+(B+C)$
  • Commutative property of addition: $A+B=B+A$ (for example: $\left(\begin{array}{cc}1& 2\end{array}\right)+\left(\begin{array}{cc}3& 4\end{array}\right)=\left(\begin{array}{cc}1+3& 2+4\end{array}\right)=\left(\begin{array}{cc}3+1& 4+2\end{array}\right)=\left(\begin{array}{cc}3& 4\end{array}\right)+\left(\begin{array}{cc}1& 2\end{array}\right)$)
  • Distributive property (multiplication by a scalar): $c(A+B)=cA+cB$

• Example: MA voting (just the questions). Each precinct might have a 2-column, 3-row matrix where the first column represents “yes” votes and the 2nd column represents “no” votes.

  	yes no
  $\begin{array}{l}Q1\\ Q2\\ Q3\end{array}$	$\left(\begin{array}{cc}1000& 800\\ 900& 700\\ 600& 500\end{array}\right)$

  ${\displaystyle \begin{array}{lll}& & \end{array}}$ ${\displaystyle }$