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.
\nb) Remember that the elements are in the form $A_{\\rm \\color{red}{ rows},\\color{blue}{columns}}$ where $A$ is the matrix.
\nFor 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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\n$A=\\var{A}$ has dimensions [[0]]$ \\times $[[1]]
\n$B=\\var{B}$ is a [[2]]$ \\times$[[3]] matrix
\n$C=\\var{C}$ is of dimension [[4]]$ \\times$[[5]]
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\n$a_{\\var{n1}\\var{m1}}=$[[0]]
\n$b_{\\var{n2}\\var{m2}}=$[[1]]
\n$c_{\\var{n3}\\var{m3}}=$[[2]]
\n$c_{\\var{n4}\\var{m4}}=$[[3]]
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\n$\\begin{pmatrix} 1&2\\\\ 4&5\\\\ \\end{pmatrix} , \\begin{pmatrix} 0&-1\\ &3\\\\2&4&-2 \\end{pmatrix}$
\nThe 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.
\nA 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).
\nWhen 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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