Matrix Problems And Solutions Pdf

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Matrices Problems And Answers. Matrix U shown below is an example of an upper triangular matrix. Also known as analytical skills interview questions, these questions will often focus on specific instances when the.

The Matrix and Solving Systems with Matrices

Usually a matrix contains numbers or algebraic expressions. You may have heard matrices called arrays , especially 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 since 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 element , and can be identified by subscripts. 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, 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 doubled 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 little trickier. Think of it like the inner dimensions have to match , and 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 multiplication , the 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 90 , Megan has a 77 , and Brittney has an See how cool this is? Oh, one more thing! Remember that multiplying matrices is not commutative order makes a difference , but 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. With a 2 by 2 matrix , you start with the upper left corner , multiply diagonally down, and then subtract the product where you multiply down diagonally from the upper right corner.

This stores the first matrix in [A]. With a 3 by 3 matrix , there are a few ways to get the determinant. First, you can use determinants of 2 by 2 matrices:. For the middle term, you have to subtract. There are other ways to get determinants of 3 by 3 matrices; this is the way I prefer to do it:.

Repeat the first two columns on the outside to the right of the matrix. Then, starting with the upper left corner , multiply diagonally down and add those three products moving to the right.

Then, starting back from the upper right corner , multiply diagonally down and subtract those three products moving to the left. Note that when the determinant is 0 , the reciprocal is undefined; therefore, there is no inverse matrix.

Why are we doing all this crazy math? Hit to get the inverse of [A]. A little easier, right? Watch the order when we multiply by the inverse matrix multiplication is not commutative , and 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 of 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:.

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 the 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 provided the following nutrition information:. Her first mixture, Mixture 1, consists of 6 cups of almonds, 3 cups of cashews, and 1 cup of pecans. Her second mixture, Mixture 2, consists of 3 cups of almonds, 6 cups of cashews, and 1 cup of pecans. Her third mixture, Mixture 3, consists of 3 cups of almonds, 1 cup of cashews, 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 matrices 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, 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 be 35 healthy males, 59 sick males, and 86 carrier males, 43 healthy females, 72 sick females, and 95 carrier females.

Pretty clever! The first table below show the points awarded by judges at a state fair for a crafts contest for Brielle, Brynn, and Briana. The second table shows the multiplier used for the degree of difficulty for each of the pieces the 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? The numbers are 5 , 7 , and 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, such as in the following problem:.

In other words, of the value of energy produced x for energy, y for manufacturing , 40 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.

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Usually a matrix contains numbers or algebraic expressions. You may have heard matrices called arrays , especially 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 since we have data being stored in different directions in a grid.

Knopoff; A matrix method for elastic wave problems. Bulletin of the Seismological Society of America ;; 54 1 : — The solution to problems of elastic wave propagation in multilayered media, in which each layer is homogeneous and where the ensemble of layers has physical properties that vary only with one coordinate, may be given as the quotient of products of matrices. In the case of SH waves, the matrices are of order two; in the case of P-SV waves the matrices are of order four. The individual matrix elements are themselves determinants of order two or four in the two cases.


b) Inversion of matrices and solution of systems of linear equations. This seems to offer almost an ideal solution of the problem of matrix multipli- cation except.


The Matrix and Solving Systems with Matrices

Examples and questions on matrices along with their solutions are presented. Example 1 The following matrix has 3 rows and 6 columns. Free Mathematics Tutorials. Definition of a Matrix The following are examples of matrices plural of matrix. Each number in a given matrix is called an element or entry.

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Examples and questions on matrices along with their solutions are presented. Example 1 The following matrix has 3 rows and 6 columns. Free Mathematics Tutorials.

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One of our mentor will revert to you within 48 hours. Meanwhile you can Enjoy the Free Study Material. Matrices and Determinants is a very important topic in Mathematics. Get Eduncle's study notes with formulas, questions and solutions to know what are Matrix and Determinant and how to solve these questions. In this brain-friendly guide, you'll study and quickly grasp the following concepts:. Introduction - What are Matrix and Determinants? Matrix and Determinant All Formulas.

Use matrix inversion method to solve the problem. Matrix word problems. Add or subtract two or three matrices in a worksheet. Exercises 26 4. Multiplication of Matrices. Always keep your workbook handy.

Examples and questions on matrices along with their solutions are presented. Example 1 The following matrix has 3 rows and 6 columns. Free Mathematics Tutorials.

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  1. PГ­o C.

    Calculus for Engineers II - Sample problems on Matrices. Solutions: Exerscise The simplest way to compute this determinant is by expanding around a.

  2. Aylin P.

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    Write the augmented matrix for the system: 3x + 2y + z = 0. 2x z = 3. Solution Matrix subtraction problems can be rewritten as matrix addition problems.

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