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Inverse Matrices and Elementary Matrices

# Inverse Matrices and Elementary Matrices - Matrices In the...

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Matrices In the previous section we used augmented matrices to denote a system of linear equations. In this section we’re going to start looking at matrices in more generality. A matrix is nothing more than a rectangular array of numbers and each of the numbers in the matrix is called an entry . Here are some examples of matrices. The size of a matrix with n rows and m columns is denoted by . In denoting the size of a matrix we always list the number of rows first and the number of columns second. Example 1 Give the size of each of the matrices above. Solution

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In this matrix the number of rows is equal to the number of columns. Matrices that have the same number of rows as columns are called square matrices . This matrix has a single column and is often called a column matrix . This matrix has a single row and is often called a row matrix .
Often when dealing with matrices we will drop the surrounding brackets and just write -2. Note that sometimes column matrices and row matrices are called column vectors and row vectors respectively. We do need to be careful with the word vector however as in later chapters the word vector will be used to denote something much more general than a column or row matrix. Because of this we will, for the most part, be using the terms column matrix and row matrix when needed instead of the column vector and row vector. There are a lot of notational issues that we’re going to have to get used to in this class. First, upper case letters are generally used to refer to matrices while lower case letters generally are used to refer to numbers. These are general rules, but as you’ll see shortly there are exceptions to them, although it will usually be easy to identify those exceptions when they happen. We will often need to refer to specific entries in a matrix and so we’ll need a notation to take care of that. The entry in the i th row and j th column of the matrix A is denoted by, In the first notation the lower case letter we use to denote the entries of a matrix will always match with the upper case letter we use to denote the matrix. So the entries of the matrix B will be denoted by . In both of these notations the first (left most) subscript will always give the row the entry is in and the second (right most) subscript will always give the column the entry is in. So, will be the entry in the 4 th row and 9 th column of C (which is assumed to be a matrix since it’s an upper case letter…). Using the lower case notation we can denote a general matrix, A , as follows,

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We don’t generally subscript the size of the matrix as we did in the second case, but on occasion it may be useful to make the size clear and in those cases we tend to subscript it as shown in the second case.
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