Chapter 4

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 that such rectangulararrays of numbers are important from many other different points of view. Numbers arealso called scalars. In general, scalars are just elements of some field.

A matrix is a rectangular array of numbers from a field F. For example, here is a matrix. 1 2 3 45 2 8 76 −9 1 2

This matrix is a 3×4 matrix because there are three rows and four columns. The columnsstand upright and are listed in order from left to right. The columns are horizontal andlisted in order from top to bottom. The convention in dealing with matrices is to alwayslist the rows first and then the columns. Also, you can remember the columns are likecolumns in a Greek temple. They stand up right 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 in which thei denotes the row and the j denotes the column. Using this notation on the above matrix,a23 = 8,a32 =−9,a12 = 2, etc.

There are various operations which are done on matrices. They can sometimes beadded, multiplied by a scalar and sometimes multiplied.

Definition 4.0.1 Let A = (ai j) and B = (bi j) be two m×n matrices. Then A+B =C whereC = (ci j) for ci j = ai j+bi j. Also if x is a scalar, xA=C where the i jth entry of C is ci j = xai jwhere the i jth entry of A is ai j. In short, ci j = xai j. The number Ai j will also typically referto the i jth entry of the matrix A. The zero matrix, denoted by 0 will be the matrix consistingof all zeros.

Do not be upset by the use of the subscripts, i j. The expression ci j = ai j + bi j is justsaying that you add corresponding entries to get the result of summing two matrices asdiscussed above.

Note that there are 2× 3 zero matrices, 3× 4 zero matrices, etc. In fact for every sizethere is a zero matrix.

With this definition, the following properties are all obvious but you should verify allof these properties are valid for A, B, and C, m×n matrices and 0 an m×n zero matrix.

A+B = B+A, (4.1)

the commutative law of addition,

(A+B)+C = A+(B+C) , (4.2)

the associative law for addition,A+0 = A, (4.3)

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