380 CHAPTER 18. LINEAR FUNCTIONS

Proof: From the definition of multiplication of matrices, (AB)ik = ∑r AirBrk. However,

bk =

 B1k...

Bnk

and so, from the way we multiply a matrix times a vector,

(Abk)i = ∑r

Air (bk)r = ∑r

AirBrk

Thus, the ith entry from the top of Abk is the ith entry in the kth column of AB showing thatindeed the claim is true.

18.2 Row Operations and Linear EquationsIn Junior High, you learned to solve things like ax = b when a ̸= 0. The fundamentalproblem considered in this section is the higher dimensional version of this Ax= b whereA is an m×n matrix. First of all, there might not even be a solution to this. Consider(

1 21 2

)(xy

)=

(13

)

Obviously there is no solution because on the left you get(

x+2yx+2y

)and you can’t have

x+ 2y equal to both 1 and 3. In contrast to religion and liberal arts, we do not toleratecontradictory assertions in mathematics. When two or more equations result in such con-tradictions, we say the equations are inconsistent. When something like this happens, wesay the solution is /0 the empty set. So how do you go about solving such equations whenthey can be solved or determining that there is no solution like the above? This involvesthe concept of a row operation.

Definition 18.2.1 The row operations applied to a matrix A consist of the following

1. Switch two rows.

2. Multiply a row by a nonzero number.

3. Replace a row by a multiple of another row added to it.

It is very useful to show that each of these row operations can be accomplished by mul-tiplication on the left by a suitable matrix called an elementary matrix. First is a definitionof the identity matrix.

Definition 18.2.2 An n× n matrix I is called the identity matrix if Ii j = 1 if i = jand Ii j = 0 if i ̸= j.

The importance of the identity matrix is that when you multiply by it, nothing changes.It acts like 1.

Proposition 18.2.3 Let A be an m× n matrix then if I is the m×m identity matrix, itfollows that IA = A and if I is the n×n identity matrix, then AI = A.