## A Beginner's Guide for Becoming a Better Problem Solver

The Primary Problem Solving Method. There are many problem-solving techniques and strategies that we could present here. However, there is essentially only one primary problem-solving method that will help you to structure and break down a problem step-by-step from the beginning to the very narrowxuwir.cf: Adam Sicinski. Matrix Diagram. Step 1: Explore the diagram below - it's a visualization of our entire Problem-Solving Matrix.. Helpful Hint: If you're looking at this graphic for the first time, we don't expect you to completely understand it at first glance. Here’s a problem from the Systems of Linear Equations and Word Problems Section; we can see how much easier it is to solve with a matrix. A florist is making 5 identical bridesmaid bouquets for a wedding.

## Solving Matrix Equations

Usually a matrix contains numbers or algebraic expressions. You may have heard matrices called arraysespecially in computer science. As an example, if you had three sisters, and you wanted an easy way to store their age and number of pairs of shoes, you could store this information in a matrix. The actual matrix is inside and includes the brackets:.

Matrices are called multi-dimensional **problem solving matrix** we have data being stored in different directions in a grid. You always go down first, and then over to get the dimensions of the matrix. Each number or variable inside the matrix is called an entry or elementand can be identified by subscripts, **problem solving matrix**. You want to keep track of how many different types of books and magazines you read, and store that information in matrices.

Here is that information, *problem solving matrix*, and how it would look in matrix form:. Thus we could see that we read 6 paper fiction, 9 online fiction, 6 paper non-fiction, 5 online non-fiction books, and 13 paper and 14 online magazines.

We could also subtract matrices this same way. If we wanted to see how many book and magazines we would have read in August if we had **problem solving matrix** what we actually read, we could multiply the August matrix by the number 2.

This is called matrix scalar multiplication ; a scalar is just a single number that we multiply with every entry. Multiplying matrices is a *problem solving matrix* trickier. Think of it like the inner dimensions have to matchand the resulting dimensions of the new matrix are the outer dimensions. To do this, you have to multiply in the following way:. Just remember when you put matrices together with matrix multiplicationthe columns what you see across on the first matrix have to correspond to the rows down on the second matrix.

You should end up with entries that correspond with the entries of each row in the first matrix. For example, with the problem above, the columns of the first matrix each had something to do with Tests, Projects, Homework, and Quizzes grades. The row down on the second matrix each had something to do with the same four items weights of grades.

But then we ended up with information on the three girls rows down on the first matrix. Alexandra has a 90Megan has a 77and *Problem solving matrix* has an See how cool this is? Oh, one more thing! Remember that multiplying matrices is not commutative order makes a differencebut is associative you can change grouping of matrices when you multiply them.

The TI graphing calculator is great for matrix operations! Here are some basic steps for storing, multiplying, adding, and subtracting matrices:. Soon we will be solving Systems of Equations using matrices, but we need to learn a few mechanics first! The inverse of a matrix is what we multiply that square matrix by to get the identity matrix.

Why are we doing all this crazy math? A little easier, right? Watch the order when we multiply by the inverse matrix multiplication is not commutativeand thank goodness for the calculator!

Watch order! It works! OK, now for the fun and easy part! Note that, like the other systems, we can do this for any system where we have the same numbers of equations as unknowns. These equations are called independent or consistent. Systems that have an infinite number of solution s called dependen t or coincident will have two equations that are basically the same.

One row of the coefficient matrix and the corresponding constant matrix is a multiple *problem solving matrix* another row. Systems with no solutions called inconsistent will have one row of the coefficient matrix a multiple of another, but the coefficient matrix will not have this.

This is called a singular matrix and the calculator will tell you so:, **problem solving matrix**. Matrices can be used for many applications, including combining data, finding areas, and solving systems. Here are some examples of those applications. In your Geometry class, you may learn a neat trick where we can get the area of a triangle using the determinant of a matrix. The formula for the area of the triangle bounded by those points is:. The store wants to know how much their inventory is worth for all the shoes.

How should we set up the matrix multiplication to determine this *problem solving matrix* best way? This way our dimension will line up. Our matrix multiplication will look like this, even though our tables look a little different I did this on a calculator :. A nut distributor wants to know the nutritional content of various mixtures of almonds, cashews, and pecans.

Her supplier has **problem solving matrix** the following nutrition information:. Her first mixture, a protein blend, **problem solving matrix**, consists *problem solving matrix* 6 cups of almonds, 3 cups of cashews, and 1 cup of pecans.

Her second mixture, a low fat mix, consists of 3 cups of almonds, 6 cups of cashews, and 1 cup of pecans. Her third mixture, a low carb mix consists of 3 cups of almonds, 1 cup of cashews, *problem solving matrix*, and 6 cups of pecans. Determine the amount of protein, carbs, and fats in a 1 cup serving of each of the mixtures. Sometimes we can just put the information we have into *problem solving matrix* to sort of see what we are going to do from there.

It makes sense to put the first group of data into a matrix with Almonds, Cashews, and Pecans as columns, and then put the second group of data into a matrix with information about Almonds, Cashews, and Pecans as rows.

This way the columns of the first matrix lines up with the rows of the second matrix, and we can perform matrix multiplication. To get the answers, we have to divide each answer by 10 to get grams per cup. The numbers in bold are our answers:. Here is one:. An outbreak of Chicken Pox hit the local public schools. There are male juniors, 80 male seniors, female juniors, and female seniors.

We can tell that this looks like matrix multiplication. And since we want to end up with a matrix that has males and females by healthy, sick and carriers, we know it will be either a 2 x 3 or a 3 x 2. But since we know that we have both juniors and seniors with males and females, *problem solving matrix*, the first matrix will probably be a 2 x 2.

Also notice that if we add up the number of students in the first matrix and the last matrix, we come up with There will **problem solving matrix** 35 healthy males, 59 sick males, and 86 carrier males, 43 healthy females, 72 sick females, and 95 carrier females.

Pretty clever! **Problem solving matrix** first table below show the points awarded by judges at a state fair for a crafts contest for Brielle, Brynn, and Briana, **problem solving matrix**. The second table shows the multiplier used for the degree of difficulty for each of the pieces **problem solving matrix** girls created.

Find the total score for each of the girls in this contest. Solve these word problems with a system of equations. Write the system, the matrix equations, and solve:. The sum of three numbers is The third number is twice the second, and is also 1 less than 3 times the first.

What are the three numbers? So the numbers are 57and Much easier than figuring it out by hand! A florist is making 5 identical bridesmaid bouquets for a wedding. She wants to have twice as many roses as the other 2 flowers combined in each bouquet. How many roses, tulips, and lilies are in each bouquet?

An application of matrices is used in this input-output analysis, which was first proposed by Wassily Leontief; in fact he won the Nobel Prize in economics in for this work.

We can express the amounts proportions the industries consume in matrices, *problem solving matrix*, such as in the following problem:. In other words, of the value of energy produced x for energy, y for manufacturing40 percent of it, or. Of the value of the manufacturing produced. The inputs are the amount used in production, and the outputs are the amounts produced. Then we do the same for manufacturing. Skip to content.

### Matrix Calculator - Symbolab

Free matrix calculator - solve matrix operations and functions step-by-step. Matrix Diagram. Step 1: Explore the diagram below - it's a visualization of our entire Problem-Solving Matrix.. Helpful Hint: If you're looking at this graphic for the first time, we don't expect you to completely understand it at first glance. Here’s a problem from the Systems of Linear Equations and Word Problems Section; we can see how much easier it is to solve with a matrix. A florist is making 5 identical bridesmaid bouquets for a wedding.