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# Basics of matrix algebra

In mathematicsa matrix plural matrices is a rectangular array or table of numberssymbolsor expressionsarranged in rows and columns. Provided that they have the same size each matrix has the same number of rows and the same number of columns as the othertwo matrices can be added or subtracted element by element see conformable matrix. The rule for matrix multiplicationhowever, is that two matrices can be multiplied only when the number of columns in the first equals the number of rows in the second i.

There is no product the other way round—a first hint that matrix multiplication is not commutative. Any matrix can be multiplied element-wise by a scalar from its associated field. In the context of abstract index notationthis ambiguously refers also to the whole matrix product. For example, the rotation of vectors in three- dimensional space is a linear transformation, which can be represented by a rotation matrix R : if v is a column vector a matrix with only one column describing the position of a point in space, the product Rv is a column vector describing the position of that point after a rotation.

The product of two transformation matrices is a matrix that represents the composition of two transformations. Another application of matrices is in the solution of systems of linear equations. If the matrix is squarethen it is possible to deduce some of its properties by computing its determinant. For example, a square matrix has an inverse if and only if its determinant is not zero.

Insight into the geometry of a linear transformation is obtainable along with other information from the matrix's eigenvalues and eigenvectors. Applications of matrices are found in most scientific fields. In computer graphicsthey are used to manipulate 3D models and project them onto a 2-dimensional screen. In probability theory and statisticsstochastic matrices are used to describe sets of probabilities; for instance, they are used within the PageRank algorithm that ranks the pages in a Google search.

Matrices are used in economics to describe systems of economic relationships. A major branch of numerical analysis is devoted to the development of efficient algorithms for matrix computations, a subject that is centuries old and is today an expanding area of research.

Matrix decomposition methods simplify computations, both theoretically and practically. Algorithms that are tailored to particular matrix structures, such as sparse matrices and near-diagonal matricesexpedite computations in finite element method and other computations.

Infinite matrices occur in planetary theory and in atomic theory. A simple example of an infinite matrix is the matrix representing the derivative operator, which acts on the Taylor series of a function.

A matrix is a rectangular array of numbers or other mathematical objects for which operations such as addition and multiplication are defined. More general types of entries are discussed below. For instance, this is a real matrix:.

The numbers, symbols, or expressions in the matrix are called its entries or its elements. The horizontal and vertical lines of entries in a matrix are called rows and columnsrespectively.Matrices are incredibly useful things that crop up in many different applied areas. For now, you'll probably only do some elementary manipulations with matrices, and then you'll move on to the next topic.

But you should not be surprised to encounter matrices again in, say, physics or engineering. The plural "matrices" is pronounced as "MAY-truh-seez". Matrices were initially based on systems of linear equations. Write down the coefficients and the answer values, including all "minus" signs. If there is "no" coefficient, then the coefficient is " 1 ". That is, given a system of linear equations, you can relate to it the matrix the grid of numbers inside the brackets which contains only the coefficients of the linear system.

This is called "an augmented matrix": the grid containing the coefficients from the left-hand side of each equation has been "augmented" with the answers from the right-hand side of each equation.

The entries of that is, the values in the matrix correspond to the x - y - and z -values in the original system, as long as the original system is arranged properly in the first place. Sometimes, you'll need to rearrange terms or insert zeroes as place-holders in your matrix. When forming the augmented matrix, use a zero for any entry where the corresponding spot in the system of linear equations is blank.

Coefficient matrices. If you form the matrix only from the coefficient values, the matrix would look like this:. This is called "the coefficient matrix". Above, we went from a linear system to an augmented matrix. You can go the other way, too. Remember that matrices require that the variables be all lined up nice and neat. And it is customary, when you have three variables, to use xyand zin that order.

So the associated linear system must be:. The Size of a matrix.

## How to Multiply Matrices

Matrices are often referred to by their sizes. The size of a matrix is given in the form of a dimension, much as a room might be referred to as "a ten-by-twelve room". The dimensions for a matrix are the rows and columns, rather than the width and length. For instance, consider the following matrix A :. The rows go side to side; the columns go up and down.

Matrix dimensions are always given with the number of rows first, followed by the number of columns. Following this convention, the following matrix B :. If the matrix has the same number of rows as columns, the matrix is said to be a "square" matrix.Matrices organizes information such as variables and constants and stores them in rows and columns, they are usually named C. Each position in a matrix is called an element.

The element 4 is in row 2 and column 2. Share on Facebook. Search Pre-Algebra All courses. All courses. Algebra 2 Equations and inequalities Overview Solve equations and simplify expressions Line plots and stem-and-leaf plots Absolute value Solve inequalities. Algebra 2 How to graph functions and linear equations Overview Functions and linear equations Graph functions and relations Graph inequalities.

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Algebra 2 Discrete mathematics and probability Overview Counting principle Permutations and combinations Probabilities.A matrix is a rectangular arrays of numbers, symbols, or expressions, arranged in rows and columns. The matrix has a long history of application in solving linear equations. Matrices can be used to compactly write and work with multiple linear equations, referred to as a system of linear equations, simultaneously.

Matrices and matrix multiplication reveal their essential features when related to linear transformations, also known as linear maps. In mathematics, a matrix plural matrices is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns.

Matrices are commonly written in box brackets.

The horizontal and vertical lines of entries in a matrix are called rows and columnsrespectively. The size of a matrix is defined by the number of rows and columns that it contains. Matrix Dimensions: Each element of a matrix is often denoted by a variable with two subscripts. The individual items numbers, symbols or expressions in a matrix are called its elements or entries.

Provided that they are the same size have the same number of rows and the same number of columnstwo matrices can be added or subtracted element by element. The rule for matrix multiplication, however, is that two matrices can be multiplied only when the number of columns in the first equals the number of rows in the second.

Any matrix can be multiplied element-wise by a scalar from its associated field. Matrices which have a single row are called row vectorsand those which have a single column are called column vectors. A matrix which has the same number of rows and columns is called a square matrix.

In some contexts, such as computer algebra programs, it is useful to consider a matrix with no rows or no columns, called an empty matrix. There are a number of operations that can be applied to modify matrices, such as matrix addition, subtraction, and scalar multiplication.

These form the basic techniques to work with matrices. These techniques can be used in calculating sums, differences and products of information such as sodas that come in three different flavors: apple, orange, and strawberry and two different packaging: bottle and can.

Two tables summarizing the total sales between last month and this month are written to illustrate the amounts. We use matrices to list data or to represent systems. Because the entries are numbers, we can perform operations on matrices. We add or subtract matrices by adding or subtracting corresponding entries. In order to do this, the entries must correspond.

Therefore, addition and subtraction of matrices is only possible when the matrices have the same dimensions. Matrix addition is commutative and is also associative, so the following is true:. Adding matrices is very simple.

Algebra 52 - An Introduction to Matrices

Just add each element in the first matrix to the corresponding element in the second matrix. You cannot add two matrices that have different dimensions.If you're seeing this message, it means we're having trouble loading external resources on our website.

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Community questions.Let us know your questions below and we'll do our best to help. Comments (11) Questions (2) Tips (0) I am one year today into a lifestyle change, gave up a HUGE addiction to sugar that had me stuffing my face with chocolate every night.

I say this because I am very disappointed at how this article handles the topic, with no science around the process of giving it up, no clarity around what makes sugar unhealthy, and spreads myths about what is "sugar".

Being "natural" means nothing. Poison Ivy is natural, doesn't mean you should eat it (I know I know hyperbole, but the point is that "natural" is just a marketing term). I am very passionate about this topic, as I have found the benefits of ending my sugar addiction to be extremely positive for my physical and mental health.

### Matrix (mathematics)

I thought sugar was making my anxiety worse, and since giving it up I've noticed a huge positive change.

I think more clearly, I'm less re-active when I feel upset, and I can focus for much longer periods of time, which really helps me at work. I'm just writing this to help point people in the right direction.

If you're interested in learning what differentiates "bad" and "good" sugar, I recommend the documentary The Sugar Film, which I've linked to below. I just hate to see people given poor health advice, when good choices from good advice can make such a huge difference in your life, help you handle everyday stresses and even large crises, by giving you confidence, strength, energy and overall health.

I've been a diabetic for almost 10 years now. The best advice i got was to cut back on starchy foods. White bread to sugar free wheat bread. I can usually hit my target sugars round 150 mgdl by lunch and less than 180 mgdl by dinner.

My wife wont let me have any fruit now in whatever form. Have been searching everywhere in the web for alternatives that dont involve sweeteners as i get really bad side effects from it, usually diarrhea.

A common misconception in weight loss is that you need to eat less, when in reality the opposite holds true.

If you restrict your calories, you will eventually slow your metabolism. I was in danger of developing Type 2 diabetes. I was advised to cut out sugar and to cut right down on carbohydrates. Any carbs eaten should be wholemeal flour. I was told this was because carbohydrates are broken down in the body into glucose (sugar) Refined flour is metabolised quickly, wholemeal is slow burn and so you don't get a big glucose spike.

I have lost a stone and a half and feel much better for it, I was thrilled to fit into a size 12 top today. It has taken 3 months to stop craving sugar now I can manage without. However, there are some great recipes in this book for families.

I've done a few of the main meal ones and we have all really enjoyed them. I agree with Karen, very confusing. Also the recommendation to use honey or maple syrup. Yes, honey and maple syrup do contain sugar but the sugar they contain is Fructose sugar- the sugars of fruits- which are not refined sugar and not considered harmful to health.

Have to say I much preferred bbcgoodfood SUGAR AND FAT FREE recipe. Carbohydrate is either a sugar, a starch (complex carbohydrate) or fibre. Can you suggest what i may have done wrong.The ratio of these two odds ratios (female over male) turns out to be the exponentiated coefficient for the interaction term of female by math: 1. Click here to report an error on this page or leave a comment Your Name (required) Your Email (must be a valid email for us to receive the report.

SERVICES Books for Loan Services and Policies Walk-In Consulting Email Consulting Fee for Service Software Purchasing and Updating Consultants for Hire Other Consulting Centers Department of Statistics Consulting Center Department of Biomathematics Consulting Clinic ABOUT US DONATE FAQ: How do I interpret odds ratios in logistic regression.

Introduction When a binary outcome variable is modeled using logistic regression, it is assumed that the logit transformation of the outcome variable has a linear relationship with the predictor variables.

From probability to odds to log of odds Everything starts with the concept of probability. We can examine the effect of a one-unit increase in math score. We can say now that the coefficient for math is the difference in the log odds. Logistic regression with multiple predictor variables and no interaction terms In general, we can have multiple predictor variables in a logistic regression model.

Logistic regression with an interaction term of two predictor variables In all the previous examples, we have said that the regression coefficient of a variable corresponds to the change in log odds and its exponentiated form corresponds to the odds ratio.

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