Applicable Mathematics/Matrices

Matrices.
A "matrix" is a rectangular array of numbers enclosed in brackets. In a notational sense, what differentiates a list of numbers from a matrix is its format. The numbers are listed so that each number has a certain, specific position between the brackets. Each number, or value, in a matrix is called an "entry".
One of the main benefits of matrices is the properties which allow them to be manipulated and used for many different, but useful purposes.
Matrices can vary in size. This variation in size is called "dimensions". Just like the dimensions of a room (width x length) matrices have dimensions (number of rows x number of columns). Thus, a 2 x 3 (read 2 by 3) matrix will have 2 rows and 3 columns.
Example of a 2 x 3 matrix:
Another term associated with matrices is "address". Like your home address, an address describes where each value, or entry, of a matrix lives. The address is composed of the lowercase letter of the matrix with the row and column number (in that order) as subscripts.
Using the 2 x 3 matrix M as an example, the positions of the values are as follows:
A "square matrix" is any matrix that has the same number of rows as it does columns.
Example: 2 x 2 or 3 x 3 matrices are both square matrices.
Take note of the numbers in red above in the 2 x 2 and 3 x 3 square matrices. These numbers are in the addresses of the main diagonal. The "main diagonal" of a square matrix is the diagonal from the upper left corner entry to the bottom right corner entry. Notice that only square matrices can have a main diagonal.
Adding and Subtracting Matrices.
To add or subtract matrices, the sum or difference is found when addition or subtraction is applied to corresponding entries.
For example,
Since addition or subtraction takes place using corresponding entries, matrices must have the same dimensions in order to complete either operation.
Consider this operation
Now consider these matrices
Properties of Equality for Matrices.
Commutative Property.
Matrix addition is commutative
formula_11
&gt;______________ formula_13
Associative Property.
Matrix addition is associative. formula_14
formula_15
formula_16
&gt;_________formula_17
Additive Identity.
The zero matrix is the "additive identity" matrix 'O'.
A "Zero matrix" is a matrix in which all of the entries are zero.
formula_18
Any matrix added to the matrix 'O' will retain its' same values.
Example of additive identity formula_19
formula_20
Additive Inverse.
It takes two matrices to form a pair of inverses. Two matrices are additive inverses if their sum is the zero matrix. This occurs when the "additive inverse" of a matrix contains the values opposite of each entry. formula_21
formula_22
Multiplicative Identity Matrix.
A "multiplicative identity matrix", usually denoted by the letter I, is any square matrix that has a value of 1 in all the entries along the main diagonal and 0 in the remaining entries.
formula_23
Any matrix multiplied by an identity matrix will retain it's original entries. formula_24
formula_25
formula_26
See Multiplying Matrices
formula_27
formula_28
Multiplicative Inverse Matrix.
If matrices formula_29 where I is the identity matrix, then A and B are multiplicative inverses of one another.
formula_30
formula_31
formula_32
Thus, matrices A and B are multiplicative inverses of each other.
Multiplying Matrices.
Back to Multiplicative Identity Matrix
Another useful property of matrices is called a "scalar". A "scalar" is a number located outside of a single matrix. To apply the scalar to the matrix, simply multiply each entry of the matrix by the "scalar".
For example,
In order to multiply two matrices together it is necessary to pay attention to their dimensions. Matrices A and J can be multiplied only if the number of columns in A equals the number of rows in J. Also, another hint is that the product of an a x j and a j x b matrix is an a x b matrix. Notice that the number of columns from the first matrix must equal the number of rows from the second matrix (j = j).
First, we'll look at how to test to see if we can multiply matrices.
Consider matrices Q and R.
formula_34
Since the number of columns of matrix Q equals the rows of matrix R, these matrices can be multiplied. This will produce a 2 x 2 matrix.
formula_35
Multiplying Q and R.
During this process, you may find your fingers and mental addition extremely helpful. You can use your left pointer finger to follow the entries in the "rows" of matrix Q and your right pointer finger to follow the "columns" of matrix R. To multiply matrices, add the products of consecutive entries in corresponding rows of matrix Q and columns of matrix R.
formula_36
formula_37
formula_38
Determinants.
Every square matrix has a value called a "determinant", and only square matrices have defined determinants. The "determinant" of a 2x2 square matrix is the difference of the products of the diagonals.
The determinant of a 2 x 2 matrix can be found as follows:
formula_39
The "down" diagonal is in red and the "up" diagonal is in blue. The up diagonals are always subtracted from the down diagonals.
Matrices that are larger than a 2 x 2 matrix become a little more complicated when finding the determinant but the same rules apply.
Let's find 
formula_40
When finding the determinant of a 3 x 3 matrix it is helpful to write the first two columns to the right side of the matrix like so,
formula_41
As shown above in the 2x2 matrix, the numbers are color coded. The blue numbers, once again, indicate they are used in the up diagonals, the red are used in the down diagonals, and those in magenta are used in both.
formula_42
formula_43
Thus, the determinant of the above 3x3 matrix is -2872.
While square matrices of any size have a determinant, there is no way to extend this diagonal method of computing that determinant for a square matrix of size 4x4 or larger.
Practice Problems.
Questions.
Match the following terms with their definitions.
1. What is a matrix? A mold in which something, such as printing type or a phonograph record, is cast or shaped.
2. How can you change one format of an equation into another? If both funds are in the same fund" family" you can do an "Exchange"
3.How do you perform scalar multiplication? You just take a regular number called a "scalar" and multiply it on every entry in the matrix.
Properties.
Match the names of the properties with their equation equivalent.
____Additive Identity&gt;______________1. formula_14
____ Additive Inverse&gt;______________2. formula_24
____ Associative Property&gt;__________3. formula_11
____ Commutative Property&gt;_________4. formula_19
____ Multiplicative Identity&gt;__________5. formula_21
____ Multiplicative Inverse&gt;__________6. formula_29
Properties Solutions
Adding Matrices.
Add the matrices.
1) formula_50
2) formula_51
3) formula_52
4) formula_53
5) formula_54
6) formula_55
Adding Matrices Solutions
Subtracting Matrices.
Subtract the matrices.
1) formula_56
2) formula_57
3) formula_58
4) formula_59
5) formula_60
Subtracting Matrices Solutions
Multiplying Matrices.
Multiply the matrices or by the scalar to find the product.
1) formula_61
2) formula_62
3) formula_63
4) formula_64
5) formula_65
Multiplying Matrices Solutions
Determinants.
Find the determinant of the following matrices.
1) formula_66
2) formula_67
3) formula_68
4) formula_69
5) formula_70
Determinants Solutions
Definitions Solutions.
Match the following terms with their definitions.
8 address&gt;_______________1. Diagonal from the upper left corner entry to the bottom right corner entry
6 determinant&gt;____________2. A rectangular array of numbers enclosed in brackets
3 dimensions&gt;____________3. Variation in size of a matrix
1 main diagonal&gt;__________4. Any matrix that has the same number of rows as it does columns
2 matrix&gt;________________5. Matrix in which all of the entries are zero
7 scalar&gt;________________6. The difference of the products of the diagonals
4 square matrix&gt;__________7. Number located outside of a single matrix which is multiplied by each entry of the matrix
5 zero matrix&gt;____________8. Describes where each value, or entry, of a matrix lives
Properties Solutions.
Match the names of the properties with their equation equivalent.
4 Additive Identity&gt;______________1. formula_14
5 Additive Inverse&gt;______________2. formula_24
1 Associative Property&gt;__________3. formula_11
3 Commutative Property&gt;_________4. formula_19
2 Multiplicative Identity&gt;__________5. formula_21
6 Multiplicative Inverse&gt;__________6. formula_29
Adding Matrices Solutions.
Add the matrices.
1) formula_77
2) formula_78
3)formula_79 Cannot be done because the matrices do not have the same dimensions.
4)formula_79 Cannot be done because the matrices do not have the same dimensions.
5) formula_81
6) formula_82
Subtracting Matrices Solutions.
Subtract the matrices.
1) formula_83
2) formula_84
3) formula_85
4) Cannot be done because the matrices do not have the same dimensions.
5) formula_86
Multiplying Matrices Solutions.
Multiply the matrices or by the scalar to find the product.
1) formula_87
2) formula_88
3) formula_89
4) formula_90
5) Cannot be done because the number of columns from the first matrix does not equal the number of rows in the second matrix.
Determinants Solutions.
Find the determinant of the following matrices.
1) 30
2) 90
3) -198
4) -33
5) 42