Introduction and Summary
This chapter introduces matrices as a way of representing data. Matrices will be used to organize data as well as to solve for variables.
The first section gives the definition of a matrix and its dimensions. It then explains how to add and subtract matrices. Not all matrices can be added to or subtracted from all other matrices, as this section explains. Matrices can be added and subtracted only if they have the same dimensions.
The second section explains two types of multiplication associated with matrices: scalar multiplication—that is, multiplication by a constant—and multiplication of two matrices. Matrix multiplication is associative, but not commutative.
Just as there is an additive identity and a multiplicative identity for all real numbers (an addition and a multiplication that does not change the number), there is an additive identity and a multiplicative identity for all matrices. The next section deals with these two identities, and introduces the identity matrix.
The subsequent section introduces operations "within" a single matrix—elementary row operations. There are three elementary row operations, and they are used to row reduce a matrix. Row reduction is used in almost all calculations with matrices, so it is important to understand this topic.
The final section of this chapter explains the concept of the inverse of a matrix. Just as most real numbers have a multiplicative inverse, most matrices also have multiplicative inverse—that is, a matrix that, when multiplied by the original matrix, yields the identity. The inverse of a matrix can be found using the row reduction, and this section explains how.
Matrices are important in Algebra II, as we will see in the next chapter. They are used in multiple ways to solve systems of equations. In addition, they are important in higher algebra. A large portion of linear algebra, which you may study in college, deals entirely with matrices. Matrices are also used by mathematicians, physicists, and biologists to organize data and study complex phenomena; for example, matrices are used to study population growth and determine when a population will stabilize.