
A rectangular array of numbers, say $n$ by $m$ , is called a matrix. The ijth element of the matrix $A$ is the element in the ith row and jth column, and is denoted as ${A}_{ij}$ .
Here are examples of matrices one two by two and the other two by three
If matrix $A$ has the same number of columns as $B$ has rows, we define the product matrix, $AB$ to be the matrix whose elements are dot products between the rows of $A$ and the columns of $B$ . The element obtained by taking the dot product of the ith row of $A$ and the jth column of $B$ is described as ${(AB)}_{ij}$ . See also Section 32.2 for a fuller discussion of matrices and their properties.
Exercises:
3.7 Find the product of the two matrices above.
3.8 Build a spreadsheet that multiplies 4 by 4 matrices. Solution
3.9 In exercise 3.8:
1. Where is the matrix product $AB$ ?
2. What appears in columns $p,q,r$ and $s$ in the first four rows?
If you change any of the entries in $A$ or $B$ the product will change automatically, so you have built an 4 by 4 matrix automatic product finder.
3. Can you use this to find the product of a 2 by 3 matrix and a 3 by 4 one? How?
4. Find the tenth power of a matrix $A$ using your product finder. (Hint: use it for $A$ and for $B$ and look in the right place and you have it.)
A vector $\stackrel{\u27f6}{v}$ can be written either as a matrix consisting of a single row, or of a single column. When writing it as a column we will write $\stackrel{\u27f6}{v}$ ; as a row, $\stackrel{\u27f6}{v}$ . The square of the length of $\stackrel{\u27f6}{v}$ can then be written as the matrix product $\stackrel{\u27f6}{v}\stackrel{\u27f6}{v}$ .
A vector $\stackrel{\u27f6}{v}$ is an eigenvector of a matrix $M$ when $M\stackrel{\u27f6}{v}$ is a multiple of $\stackrel{\u27f6}{v}$ . The multiple is called the eigenvalue of $M$ having eigenvector $\stackrel{\u27f6}{v}$ . If the eigenvalue is $s$ , then we have $M\stackrel{\u27f6}{v}=s\stackrel{\u27f6}{v}$ .
The applet here allows you to enter any 2 by 2 matrix, and move the vector $\stackrel{\u27f6}{v}$ around. When $M\stackrel{\u27f6}{v}$ lines up with $\stackrel{\u27f6}{v}$ , $\stackrel{\u27f6}{v}$ is an eigenvector of $M$ with real eigenvalue which is given by the ratio of the length of $M\stackrel{\u27f6}{v}$ (called $\stackrel{\u27f6}{v\text{'}}$ in the applet) to that of $\stackrel{\u27f6}{v}$ , with a sign that is positive when they point in the same direction.
Exercise 3.10 Choose a symmetric matrix and use the applet to determine the two eigenvectors, approximately. Draw them on a piece of paper. Can you notice something about them? What?
