Chapter 5

Matrices5.1 Matrix Arithmetic

5.1.1 Addition And Scalar Multiplication Of MatricesYou have now solved systems of equations by writing them in terms of an augmented matrixand then doing row operations on this augmented matrix. It turns out such rectangulararrays of numbers are important from many other different points of view. Numbers arealso called scalars. In this book, numbers will generally be either real or complex numbers.I will refer to the set of numbers as F sometimes when it is not important to worry aboutwhether the number is real or complex. Thus F can be either the real numbers R or thecomplex numbers C. However, most of the algebraic considerations hold for more generalfields of scalars.

A matrix is a rectangular array of numbers. Several of them are referred to as matrices.For example, here is a matrix.  1 2 3 4

5 2 8 76 −9 1 2

The size or dimension of a matrix is defined as m× n where m is the number of rows andn is the number of columns. The above matrix is a 3× 4 matrix because there are threerows and four columns. The first row is (1 2 3 4) , the second row is (5 2 8 7) and so forth.

The first column is

 156

 . When specifying the size of a matrix, you always list the

number of rows before the number of columns. Also, you can remember the columns arelike columns in a Greek temple. They stand upright while the rows just lie there like rowsmade by a tractor in a plowed field. Elements of the matrix are identified according toposition in the matrix. For example, 8 is in position 2,3 because it is in the second row andthe third column. You might remember that you always list the rows before the columnsby using the phrase Rowman Catholic. The symbol, (ai j) refers to a matrix. The entry inthe ith row and the jth column of this matrix is denoted by ai j. Using this notation on theabove matrix, a23 = 8,a32 =−9,a12 = 2, etc.

There are various operations which are done on matrices. Matrices can be added mul-tiplied by a scalar, and multiplied by other matrices. To illustrate scalar multiplication,consider the following example in which a matrix is being multiplied by the scalar 3.

3

 1 2 3 45 2 8 76 −9 1 2

=

 3 6 9 1215 6 24 2118 −27 3 6

 .

The new matrix is obtained by multiplying every entry of the original matrix by the givenscalar. If A is an m×n matrix, −A is defined to equal (−1)A.

Two matrices must be the same size to be added. The sum of two matrices is a matrix

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