We cannot simply multiply matrices the way we add and scalar multiply.

The dot product combines two matrices, not necessarily of the same dimension into a third matrix. Understand the location of each element will help you understand the location of the final dot products.

**Dimensions Determine the Possibility of a Dot Product**

First, we must determine whether the dot product of a matrix is possible.

The dimension of the columns of the first matrix needs to be identical to the dimension of the rows of the second matrix. The simplest way to determine this is either visually (as can be seen above) or writing the dimensions in order. Notice 3X3 and 3X1 both have a 3 in the middle. These matching 3’s indicate we are allowed to perform the dot product operation. The final product will the the outside dimensions, here we have a 3X1 final matrix.

The following is a visual of matching dimensions and the final product of a dot product.

**Dot Product as an Operation**

Once we determine that the dimensions create a viable situation, we must use our knowledge of the location of the elements to determine how to complete the operation. I find it easiest to trace the progress with my fingers to ensure I don’t skip any elements.

Notice, in the above example, we are determining the first row, first column element, here 58. So we must combine all the elements from the first row from the first matrix to all the elements in the first column from the second matrix. Namely,

(1*7)+(2*9)+(3*11)=7+18+33=58

We continue with this process to find each of the remaining elements in the “answer” matrix.

First row, second column in the answer will be the first row in the first matrix and the second column in the second matrix.

(1*8)+(2*10)+(3*12)=8+20+36=64

This process continues to complete the entire matrix.

Here is another example:

It is important to account for every element or you may arrive at an incorrect example. As you get better at the dot product, you will not need to show any work, however anyone new to the process is advised to show the steps until mastery is achieved.

**The Inverse of a Matrix**

Remember when we defined inverses for real numbers? The product of any number (except zero) and its multiplicative inverse is 1. For example, 2* 1/2=1. In the real number system, the multiplicative inverse in simply the reciprocal. With matrices, the inverse is a little more in depth, however its properties are the same.

The fraction in front of our resultant matrix is known as the determinant. The scalar multiplication of the determinant and the “switched” matrix gives us our inverse.

**Solving Systems of Equations Using Matrices **

Generally, matrices are useful for finding the solution to more complex systems of equations. We utilize the inverse of a matrix to quickly compute even the most difficult systems regardless of the number of variables or dimensions. This application of matrices is evident in many other fields beyond mathematics, however, we will stick to a simple system for this course to get the point across.

We can set up our matrices as shown below and use the simple process of multiplying both sides of the equation by the inverse to determine the solution set.

As you can see this process seems excessive and lengthy for a system of two linear equations, however as the systems get more complex and additional variables are added, the simplicity of matrices supersedes any algebraic approach.

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We simply add the elements in the same corresponding position and obtain a matrix with the same dimensions. In this case a 2X2.

For example:

If two matrices do not have the same dimensions, they cannot be added.

** **

Scalar multiplication is the simple process of increasing each element within a matrix by a single product.

Because we only have one number as the multiplier, specific dimensions are not necessary.

Here are two examples:

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A matrix is made up of elements within those rows and columns. Understanding placement helps the progression into operations on matrices, such as addition, subtraction, scalar multiplication, dot products, and inverses.

In the following example we see a 3×3 matrix. We label the dimensions of a matrix as rows X columns. Each number in a matrix is an element. In the general matrix, we see each element also has a corresponding position named just as the matrix was named, rows X columns.

We are able to perform operations on matrices as long as their dimensions align. Different operations require a different alignment of dimensions. You will see more of these operations in the proceeding sections.

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**Graphing Systems of Non Linear Equations to Determine Solutions**

A single intersection point can be seen when a circle and a line are tangent to each other.

One example of two intersection points is seen in the right example with the intersection of a cubic equation and a line.

Another example of two intersection points is seen in the intersection of a circle and a line, or the intersection of two circles.

One example of a system of equations with three intersection points is seen to the left with the intersection of a circle and a parabola. Notice, the vertex of the parabola must be tangent to the circle, otherwise there would be either two or four intersection points.

Finally, four intersection points can be observed with the intersection of a circle and an ellipse.

Each of these examples by no means exhausts the possibilities of intersection points. They simply illustrate the possibilities of up to four solutions to a non linear system of equations.

**Solving Systems of Non Linear Equations Algebraically**

Just as before, a more accurate way to determine common solutions to a system of equations is to algebraically solve the systems. Just as before, we can use the same tools to solve these systems: substitution and elimination. The only distinct difference is working with solving utilizing factoring and roots. These processes are not explained until Lesson 10 and Lesson 11 so be sure you understand those properties prior to continued progress in this section.

When solving a system with one quadratic equation and one linear equation, it is generally easiest to utilize substitution. This is because the variables must match up perfectly, and the presence of exponents can hinder elimination for this reason.

Lets check out an example of solving a system with a circle and a line. We can have either one or two solutions to this system.

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The important vocabulary for this concept is:

terms coefficients constants like terms

The parts of an expression that are added(or subtracted) together are called the **terms. **This expression has 4 terms, 4x, -8, y, and -3.

The number part of a term with a variable part is called the **coefficient** of the term, 4 and 1 are the coefficients in this equation.

A **constant** **term** has a number part but no variable part, such as -8 and -3 in the expression above.

**Like terms** are terms that have the same variable parts. Notice 4x and y are not like terms, because their variables are different, and all constants are like terms with each other because of the absence of a variable.

The expression below also does not have any like terms. Even though each of the variable terms contain an x, they are different types of x. The only number that will not affect the “nature” of the term in terms of like terms is the coefficient. Here, the variable (or smaller number) distinguishes theses are different types of x.

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**Complicated Expressions**

As you progress through your mathematical journey you will be faced with a growing level of complexity in the expressions and equations you encounter. Fear not because even the most difficult formulas and equations can be broken down into the basic features outlined above.

For example, the following equation is how compound interest is calculated. This equation is used for investments and payments such as monthly car payments. It may seem complicated at first, however when we see that it is simply the product of P (the principal amount, or original amount) and a factor not depending on P, in this case the rate of interest. For example, if you decide to take out a loan from a bank to buy a car, many banks have a set interest for loans based on your financial stability, not on the value of the loan.

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**Stem and Leaf Plots**

A stem and leaf plot organizes data based on their digits. For example, the data below is representative of the set

{19, 22, 25, 26, 27, 28, 29, 30, 34, 36, 37, 42, 43, 44, 46, 48, 48, 49, 52, 53, 55, 57, 58, 62}

You will notice it helps organize values by their values. You can choose any organization for these plots (including 100s, tenths, etc)

**Histograms**

Histograms show the frequency of data on intervals of equal size, with no gaps or overlaps. This is showing ages on the horizontal axis and some frequency based on age on the vertical axis.

**Box and Whisker Plots**

Box and whisker plots organize data into four groups of approximately equal size. It shows the center(median), where most values lie (within the box) and any outlying data (extreme minimums or maximums).

As you progress through mathematics and life in general it is important to understand how to read data and how to determine its viability. If you are aware of the possibility of bias you will not be swayed by inaccurate data that is trying to convince you of something that is false. Don’t be fooled!!

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1. In a **random sample**, every member of the population has an equal chance of being selected.

2. In a **stratified random sample**, the population is divided into distinct groups. Members are selected at random from each group.

3. In a **systematic sample**, a rule is used to select members of the population.

4. In a **convenience sample**, only members of the populations who are easily accessible are selected.

5. In a **self selected sample**, members of the populations select themselves by volunteering.

As you can imagine, the variety of sampling can create a level of bias. A **biased sample** is a sample that is not representative of the population. In a biased sample, parts of the populations may be over- or under-represented.

Another way for bias to appear in samples is through a **biased question**, or a question that encourages a particular response.

**Analyzing Sets of Data**

You can find values that represent a typical data value using the following measures of central tendency:

**mean**

This is the average of a numerical data set and is found by adding all values in a data set and dividing by the number of items in that set.

**median**

This is the middle number of a data set when numbers are written in numerical order. If the data set has an even number of values, the median is the mean of the two middle values.

**mode**

This is the value that occurs most frequently. There may be one mode, no mode, or more than one mode.

You can find values that describe the spread of data using the following measures of *dispersion*(or the description of the spread of data):

**range**

This is the difference of the greatest and least value.

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ABC ACB BAC BCA CAB CBA

A combination is a selection of objects in which order is not important. Using the same example above. The number of possible ways to choose 3 letters from the letters ABC is just one way:

ABC

Notice, order doesn’t matter with combinations, so we eliminate many of the options that permutations create.

There are two formulas that illustrate these ideas algebraically. You can use these when you have many more choices than 3 letters (say 6 lotto numbers where order matters… or 4, 8, 15, 16, 23, 42 )

Your calculator probably has buttons for combinations and permutations to save you from this memorization conundrum!

Before we go through an example, lets review factorials!

Factorials are literally the number multiplied by the numbers below it until you reach 1. For example,

4!=4·3·2·1=24

You can also reduce all the writing by using smaller factorials, for example:

5!=5·4·3·2·1

or

5!=5·4!

This helps reducing these funky fractions later in more advanced probability! Also remember the rule, 0!=1. You will prove this in pre calculus.

Using our first example again, lets find the permutations and combinations of those 3 letters.

Permutations of ABC

3P3=3!/(3-3)!=3!/0!=3!/1=3!=3·2·1=6 exactly what we got by reordering our letters above.

Combinations of ABC

3C3=3!/(3-3)!3!=3!/0!3!=3!/3!=1 again, this is what we saw in our original example!

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A possible result of an experiment is an *outcome*. For instance, when you toss a die, there are six possible outcomes, 1, 2, 3, 4, 5, or 6. An* event* is an outcome or collection of outcomes, such as rolling an odd number. The set of all possible outcomes is called a *sample space*.

**Theoretical Probability**

The outcomes for a specified event are called *favorable outcomes.* When all the outcomes are equally likely, the theoretical probability of the event, A, can be found using the following formula:

**Experimental Probability**

An experimental probability is based on repeated *trials* of an experiment. The number of trials is the number of times the experiment is repeated. Each trial in which a favorable outcome occurs is called a *success. *The following formula shows the probability that a successful event, A, occurs in an experiment. Note the number of trials would total the number of events labeled A (or successes) and not A (non successful events).

If you have more than one event that are occurring (A and/or B) , you can use the following formulas.

For one event OR the other(P(A or B)):

**Events A and B have no common outcomes**

(for example roll a dice then toss a coin: P(heads)+P(six). These two events will never be the same.)

**Events A and B that have at least one common outcome**

(for example, looking at girls in a particular school, P(girl) or P(basketball player) can occur at the same time, so you must subtract out the duplicate. Say Helen is a girl AND plays basketball, when you find

*P(girl or basketball player)=P(girl)+P(basketball player)*,

helen will be counted twice. Hence we adjust the equation to read,

*P(girl or basketball player)= P(girl)+P(basketball player)-P(girls who play basketball)*

just like the generic equation above.

For two events occurring at the same time, (P(A and B))

**Events A and B are independent**

An example where two events are independent would be drawing a marble from a bag and putting it back before you draw the second marble. The first choice wouldn’t affect the second, hence they would be independent.

**Events A and B are dependent**

Using the marble example again, if you drew a marble and DIDN’T put it back, it would affect the probability of the second marble being drawn. The odds that second marble would be picked would be greater because there would be less to choose from.

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Performing operations on rational expressions is similar to performing operations on numerical fractions. Any common factors in the numerator and denominator should be divided out, and the original expression should be used when finding excluded values. The following operations have specific rules that are listed below.

**Multiplication**

Notice we just multiply straight across. Don’t be tempted to do this when addition comes out to play!

**Division**

When you divide by a rational number (or function), its the same as multiplying by the inverse! Flip and multiply!

**Addition**

If you have the same denominator you can simply add the top and combine the bottom. For example, 1/2+1/2=(1+1)/2=2/2=1!

**Subtraction**

The same rule applies when you subtract. Be SURE the bottom is identical!

What happens if the denominator is not identical?! Well! I am SO glad you asked!!!

This is when you use the lowest common denominator (LCD) to create identical denominators. Then you solve just like the previous addition and subtraction rules!

**Solving Rational Equations and Functions**

You can use the following steps to solve a rational equation.

1. Rewrite the rational equation using the cross products property or by multiplying each side by the LCD of the rational expressions in the equation.

2. Solve the rewritten equation.

3. Check for extraneous solutions.

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