Introduction to Matrices (DP IB Applications & Interpretation (AI): HL): Revision Note

Introduction to matrices

What are matrices?

• A matrix is a rectangular array of elements (numerical or algebraic) that are arranged in rows and columns

• The order of a matrix is defined by the number of rows and columns that it has

  • The order of a matrix with $m$ rows and $n$ columns is $m\times n$

    • e.g. the matrix $\left(\begin{array}{cc}1& 3\\ -2& 0\\ 1& 2\end{array}\right)$ has order 3 × 2

• A matrix $\mathit{A}$ can be defined by $\mathit{A}=\left(a_{ij}\right)$ where $i=1,2,3,...,m$ and $j=1,2,3,...,n$

  • $a_{ij}$ refers to the element in row $i$ and column $j$

What type of matrices are there?

• A column matrix (or column vector) is a matrix with a single column, $n=1$

  • e.g. $\left(\begin{array}{c}1\\ 0\\ -2\end{array}\right)$

• A row matrix is a matrix with a single row, $m=1$

  • e.g. $\left(\begin{array}{ccc}1& 0& -2\end{array}\right)$

• A square matrix is one in which the number of rows is equal to the number of columns, $m=n$

  • e.g. $\left(\begin{array}{cc}1& 2\\ 0& -1\end{array}\right)$

• Two matrices are equal when they are of the same order and their corresponding elements are equal

  • $\mathit{A}=\mathit{B}$ if $a_{ij}=b_{ij}$for all elements

• A zero matrix, $\mathit{O}$, is a matrix in which all the elements are $0$

  • e.g. $\mathit{O}=\left(\begin{array}{cc}0& 0\\ 0& 0\end{array}\right)$

• The identity matrix, $\mathit{I}$, is a square matrix in which all elements along the leading diagonal are $1$ and the rest are $0$

  • e.g. $\mathit{I}=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$