Period+6-Properties+of+Matrices

Alex Volynsky & Caitlin Bakofsky

**General Summary**

**A matrix is simply defined as an ordered set of numbers, arranged in a rectangular __form__, either representing a large set of data in a very concise manner, or the coefficients of a system. ** A matrix can not only be a rectangular array of numbers, but also a rectangular array of symbols or expressions.

** The Size of a Matrix **

**NOTES **

Matrices are typically referred to by their varying sizes. The capacity and size of a matrix are provided in the form of a dimension, similarly to how a room might be known as "a ten-by-twelve room". However, the dimensions for a matrix are not width and length, but rather rows and columns. For example, examine the succeeding matrix:



Due to the fact that the matrix contains three rows  and four columns , the size would be stated as 3 × 4, and pronounced as “three-by-four.”



As everybody should already know by now, rows go side to side while the columns go up and down. "Row" and "column" are technical terms, and of course cannot be utilized interchangeably. As aforementioned, matrix dimensions must be given by the total number of rows first, and then the total number of columns. Using this rule, take a look at the next matrix:

This is obviously a 2 × 3, due to the fact that it has 2 rows and then 3 columns.

**EXAMPLE PROBLEMS **

Now try some on your own! What do you think the dimensions are of these matrices? 1. 2. 3.

<span style="background-color: transparent; color: #000000; display: block; font-family: Arial; font-size: 13px; text-decoration: none; vertical-align: baseline;">Answer Key: 1. (4 X 4 << This is a square matrix! You will learn about this a little later.) 2. (3 X 1) 3. (1 X 3) <span style="background-color: transparent; color: #000000; display: block; font-family: Arial; font-size: 13px; text-decoration: none; vertical-align: baseline;">For more information and extra practice, take a quick look at Period 8’s page here. **<span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 13px; text-decoration: none; vertical-align: baseline;">Web bibliography: **<span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 13px; text-decoration: none; vertical-align: baseline;">-http://www.purplemath.com/modules/matrices.htm ** The Square Matrix **

**<span style="background-color: transparent; color: #000000; display: block; font-family: Arial; font-size: 13px; text-decoration: none; vertical-align: baseline;">NOTES ** <span style="background-color: transparent; color: #000000; display: block; font-family: Arial; font-size: 13px; text-align: left; text-decoration: none; vertical-align: baseline;">If the matrix has the same number of rows as columns, such as the following example, the matrix is said to be a "square" matrix. An example would be the following:

<span style="background-color: transparent; color: #000000; display: block; font-family: Arial; font-size: 13px; text-align: left; text-decoration: none; vertical-align: baseline;">This is obviously a 3 x 3 matrix; it has 3 columns and 3 rows, meaning that it is in fact a square matrix.

**<span style="background-color: transparent; color: #000000; display: inline ! important; font-family: Arial; font-size: 13px; text-align: left; text-decoration: none; vertical-align: baseline;">EXAMPLE PROBLEMS **

<span style="background-color: transparent; color: #000000; display: block; font-family: Arial; font-size: 13px; text-align: left; text-decoration: none; vertical-align: baseline;">Now try some on your own! Write whether or not the following matrices are square matrices.

<span style="background-color: transparent; display: block; font-family: Times; font-size: 13px; text-align: center;">1. 2. 3.

<span style="background-color: transparent; color: #000000; display: block; font-family: Arial; font-size: 13px; text-align: left; text-decoration: none; vertical-align: baseline;">Answer Key: 1. (This is a 3 x 3 matrix, just like the one in the example. Because it has 3 rows and 3 columns it is definitely a square matrix.) 2. (This is a 3 x 2 matrix. Because it has 3 rows but only 2 columns, it is not a square matrix.) 3. (This is a 2 x 2 matrix, meaning that it has 2 rows and 2 columns. It is also a square matrix.)

<span style="background-color: transparent; color: #000000; display: block; font-family: Arial; font-size: 13px; text-align: left; text-decoration: none; vertical-align: baseline;">For some more practice and information, try and check out Period 8's page here, and scroll down a little bit. <span style="background-color: transparent; color: #000000; display: block; font-family: Arial; font-size: 13px; text-align: left; text-decoration: none; vertical-align: baseline;">You can also visit visit the page on Cramer's rule involving square matrices here and here, and how to __apply__ your knowledge of square matrices to linear systems here and here (link will be added when the group makes their website). ** Web bibliography: **<span style="display: inline ! important; font-size: 13px; text-align: left;"> - http://home.scarlet.be/math/matr.htm#Square-matrix -http://www.purplemath.com/modules/matrices.htm

** The Column and Row Matrix **

** NOTES ** Simply put, a column matrix is a matrix that only has one column, while a row matrix is a matrix that only has one row. The following is an example of a column matrix:



This matrix is a 3 x 1 matrix, meaning that it has 3 rows but only 1 column, which fits the criteria for a column matrix. Take this example of a row matrix:



** EXAMPLE PROBLEMS ** Now try it out on your own. State the dimensions of the next few matrices, and say whether they are either column, row, or even square matrices:

1.

2.  3.

Answer Key: 1. (The first matrix is a 1 x 3 matrix, meaning that it has 1 row and 3 columns. It is definitely a row matrix.) 2. (The second matrix is a 2 x 2 matrix, with 2 rows and 2 columns, making it a square matrix.) 3. (The third is a 3 x 2 matrix, with 3 rows and 2 columns. It is neither a column, nor row, nor square matrix.)

<span style="background-color: transparent; color: #000000; display: block; font-family: Arial; font-size: 13px; text-align: left; text-decoration: none; vertical-align: baseline;">For some more practice and information on column and row matrices, check out Period 8's page here and scroll down a little bit. <span style="background-color: transparent; color: #000000; display: block; font-family: Arial; font-size: 13px; text-align: left; text-decoration: none; vertical-align: baseline;">Also check out how column and row matrices can be used in arithmetic and operations, here! **<span style="background-color: transparent; color: #000000; display: block; font-family: Arial; font-size: 13px; text-align: left; text-decoration: none; vertical-align: baseline;">Web bibliography: ** -http://home.scarlet.be/math/matr.htm#Row-matrix <span style="background-color: transparent; color: #000000; display: block; font-family: Arial; font-size: 13px; text-align: left; text-decoration: none; vertical-align: baseline;">-http://home.scarlet.be/math/matr.htm#Column-matrix - http://www.purplemath.com/modules/matrices2.htm

** The Identity Matrix **

**NOTES** In order to completely understand an identity matrix, you must first learn about what a diagonal matrix is. A diagonal matrix is a square matrix where all of the non-diagonal elements are equal to 0, and the matrix itself is defined by its diagonal elements. Study the succeeding example:

This is a diagonal matrix, due to the fact that all of the non-diagonal elements are equal to 0. This matrix in particular would be referred to by "diag (7,5,6)." The name of the matrix is dependent on the diagonal values, and the amount of rows/columns. An identity matrix in particular is where all of the diagonal elements are equal to 1. Scrutinize this chain of examples:

... and so on All of the diagonal elements are clearly equal to one, which fits the criteria of an identity matrix.

**EXAMPLE PROBLEMS** Now try it out on your own! Identify what type of matrix the problem is, and then write out its denotation: 1.  2.   3.   Answer Key: 1. (This is not only a 5 x 5 square matrix, but also an identity matrix because all of the diagonal elements add up to 1.) 2. (This is a 3 x 1 column matrix, because it only has 1 column.) 2. (This is a 3 x 3 matrix, because it has 3 rows and then 3 columns, making it a square matrix.)

Take a look at Period 8's website for more practice/help with this concept here. Scroll all the way to the bottom. The reason why the identity matrix got its name will be uncovered when you learn about multiplying matrices. Click here and scroll down a little for more information. Cramer's rule for a 2 x 2 system also involves multiplying diagonally. Click here to explore this concept as well.

**Web bibliography:** -http://home.scarlet.be/math/matr.htm#Diagonal-matrix -http://home.scarlet.be/math/matr.htm#An-identity-matrix-I -http://www.purplemath.com/modules/matrices2.htm