Introduction to Matrices

A matrix (plural: matrices) is simplya rectangular array of”things”. For now, we’llassumethe”things”are numbers, but as you goon in mathematics, you’llfindthat matrices can be arrays of very general objects. Pretty much all that’s required is that you be able to add, subtract, and multiply the”things”. Here are some examples of matrices. Notice that it is sometimes useful to have variables as entries, as long as the variables represent the same sorts of”things”as appear in the other slots. In our examples, we’llalwaysassume that all the slots are filled with numbers. All our examples contain only real numbers, but matrices of complex numbers are very common.

The first example is a square matrix; the next isa matrix (2 rows and 4 columns—if we talk about a matrix that is” “wemeanithasrows and columns). The final two examples consist of a single column matrix, and a single row matrix. These final two examples are often called”vectors”—the first is called a”column vector”and the second, a”row vector”. We’lluseonlycolumn vectors in this introduction. Often we are interested in representinga general matrix with variables in every location, and that is usually done as follows: . . . . . . . . . . . . . . . 

The number inro w and column is represented by , where and . Sometimes when there is no question about the dimensions of a matrix, the entire matrix can simply be referred to as: 1.1 Additionand Subtraction of Matrices As long as you can add and subtract the”things”in your matrices, you can add and subtract the matricesthemselv es. The addition and subtraction occurs in the obvious way—element by element. Here area couple of examples: To find what goes inro wand column of the sum or difference, just add or subtract the entries in row and column of the matrices being added or subtracted. In order to make sense, both of the matrices in the sum or difference must have the same number of rows and columns. It makes no sense, for example, to adda matrix to a matrix. 1.2 Multiplication of Matrices When you add or subtract matrices, the two matrices that you add or subtract must have the same number of rows and the same number of columns. In other words, both must have the same shape. For matrix multiplication, all that is required is that the number of columns of the first matrix be the same as the number of rows of the second matrix. In other words, you can multiply an matrix by a matrix, with the matrix on the left and the matrix on the right.

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