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a)      Remember that the dimensions are $\\rm \\color{red}{rows} \\times \\color{blue}{columns}$, so you need to count the number or rwos and columns in the matrix and wrtite them in that order.

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b)      Remember that the elements are in the form $A_{\\rm \\color{red}{ rows},\\color{blue}{columns}}$ where $A$ is the matrix.

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For example if we are looking for $a_{12}$ we look at the matrix $A= \\begin{pmatrix} \\var{a11} &\\bf( \\underline{\\var{a12}}) \\\\ \\var{a21} & \\var{a22}\\end{pmatrix}$ we want the the element on $\\rm \\color{red}{row ~ 1}$ and $\\rm \\color{blue}{column ~ 2}$ which in this case is $\\bf \\underline{\\var{a12}}$

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Example 1 -- Give the dimensions of the following matrices:

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$A=\\var{A}$ has dimensions [[0]]$\\times$[[1]]

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$B=\\var{B}$ is a [[2]]$\\times$[[3]] matrix

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$C=\\var{C}$ is of dimension [[4]]$\\times$[[5]]

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Example 2 -- Give the values of the following elements of the matrices from example 1 above:

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$a_{\\var{n1}\\var{m1}}=$[[0]]

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$b_{\\var{n2}\\var{m2}}=$[[1]]

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$c_{\\var{n3}\\var{m3}}=$[[2]]

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$c_{\\var{n4}\\var{m4}}=$[[3]]

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Matrices are rectangular arrays of numbers arranged in rows and columns.  For example

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$\\begin{pmatrix} 1&2\\\\ 4&5\\\\ \\end{pmatrix} , \\begin{pmatrix} 0&-1\\ &3\\\\2&4&-2 \\end{pmatrix}$

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The rows run across the matrices while the columns run down the matrices.  Thus in the first matrix above the numbers $\\begin{pmatrix}1&2\\end{pmatrix}$ are in the first row while the numbers $\\begin{pmatrix}4&5\\end{pmatrix}$ are in the second row and similarly the numbers $\\begin{pmatrix} 1\\\\ 4\\\\ \\end{pmatrix}$ are in the first column while the numbers $\\begin{pmatrix} 2\\\\ 5\\\\ \\end{pmatrix}$ are in the second column.

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A matrix with $\\bf m$ rows and $\\bf n$ columns is called a matrix of order $\\bf m\\times n$ or dimension $\\bf m\\times n$ (or an $\\bf m\\times n$ matrix for brevity).

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When working with matrices the positions of the numbers in the arrays are as important as the actual values of the numbers.  Given a matrix called  $A$ the number in row $i$ and column $j$ is usually denoted $a_{ij}$ it is also sometimes called the $ij^{th}$ element of the matrix $A$.

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aij  notation and definition of the order of a matrix.

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