My thanks to an alert reader for asking, “*What are the practical use of matrices in day to day life?*” The most direct answer is, “*It depends on your own day to day life*.” Let’s consider some practical uses of matrix mathematics in a variety of settings, along with a brief introduction to matrices.

## Applications of Matrix Mathematics

Matrix mathematics applies to several branches of science, as well as different mathematical disciplines. Let’s start with computer graphics, then touch on science, and return to mathematics.

We see the results of matrix mathematics in every computer-generated image that has a reflection, or distortion effects such as light passing through rippling water.

Before computer graphics, the science of optics used matrix mathematics to account for reflection and for refraction.

Matrix arithmetic helps us calculate the electrical properties of a circuit, with voltage, amperage, resistance, etc.

In mathematics, one application of matrix notation supports graph theory. In an adjacency matrix, the integer values of each element indicates how many connections a particular node has.

The field of probability and statistics may use matrix representations. A probability vector lists the probabilities of different outcomes of one trial. A stochastic matrix is a square matrix whose rows are probability vectors. Computers run Markov simulations based on stochastic matrices in order to model events ranging from gambling through weather forecasting to quantum mechanics.

Matrix mathematics simplifies linear algebra, at least in providing a more compact way to deal with groups of equations in linear algebra.

## Introduction to Matrix Arithmetic

A matrix organizes a group of numbers, or variables, with specific rules of arithmetic. It is represented as a rectangular group of rows and columns, such as . This “2X3” matrix has two rows and three columns; the number ’23’ is in the second row of the third column.

An example of a square matrix with variables, rather than numbers, is . This is a square matrix because the number of rows equals the number of columns.

We can only add matrices of the same dimensions, because we add the corresponding elements. .

Matrix multiplication is another matter entirely. Let’s multiply matrices MP=R. M is an mXn matrix; P is nXp; and the result R will have dimension mXp. Note that the number of columns of the left-hand matrix, M, must equal the number of rows of the right hand matrix, P. For example:

.

A matrix can also multiply, or be multiplied by, a vector.

## Graphic Uses of Matrix Mathematics

Graphic software uses matrix mathematics to process linear transformations to render images. A square matrix, one with exactly as many rows as columns, can represent a linear transformation of a geometric object. For example, in the Cartesian X-Y plane, the matrix reflects an object in the vertical Y axis. In a video game, this would render the upside-down mirror image of a castle reflected in a lake.

If the video game has curved reflecting surfaces, such as a shiny silver goblet, the linear transformation matrix would be more complicated, to stretch or shrink the reflection.

## The Identity Matrix and the Inverse Matrix

The Identity matrix is an nXn square matrix with ones on the diagonal and zeroes elsewhere. It causes absolutely no change as a linear transformation; much like multiplying an ordinary number by one. The dimension of an Identity matrix is shown by a subscript, so I_{2} = is the 2X2 Identity matrix.

Suppose we have two square nXn matrices, A and B, such that AB=I_{n}. Then we call B the inverse matrix of A, and show it as A^{-1}. The first practical point is that the inverse matrix A^{-1} reverses the changes made by the original linear transformation matrix A.

## The Determinant

Another important task in matrix arithmetic is to calculate the determinant of a 2X2 square matrix. For matrix M= , the determinant is |M| = a*d – b*c.

If the determinant of M is zero, then *no* inverse matrix M^{-1} exists.

On the other hand, if we apply M as the linear transformation of a unit square U into U_{M}, then the determinant |M| is the area of that transformed square. In a sense, the determinant is the size, or “norm”, of a square matrix.

## Daily Matrix Applications

Matrix mathematics has many applications. Mathematicians, scientists and engineers represent groups of equations as matrices; then they have a systematic way of doing the math. Computers have embedded matrix arithmetic in graphic processing algorithms, especially to render reflection and refraction. Some properties of matrix mathematics are important in math theory.

However, few of us are likely to consciously apply matrix mathematics in our day to day lives.

Readers, please leave a comment: how do you use matrices on a daily basis?

© Copyright 2013 Mike DeHaan, All rights Reserved. Written For: Decoded Science
PARAS JAIN says

Really helpful to me !!!

Sasmitha says

Please clearly tell what are the applications in various fields

M I Shamsi says

what is the practical meaning of determinate of a Matrix? If a Matrix A determinate is 10, what actually this 10 means?

onesmo john says

application of matrix its helpfully in encrypting message in data communication especcialy inverse of matrix

Nima says

May I request that how civil engineer can apply the knowledge of matrics in field?

S.meena says

It’s very useful.thanks.

Derrick says

Thanks

syed Junaid ul Hassan says

how we can learn applications of matrices… mean that how can apply

Mike Smolen says

You mention in about the second page that matrices are known from weather studies. Can you point me to such citations? I’m working on weather data and would love to see what others have done and see if it would be useful for my application.

Lavanya says

Pls clearly mention what is tha application of matrices in day to day life

May phuu says

What can I solve matrix sin cos with definition of inverse of matrix?

Dilkhush kumar says

sir i really want to know what is application of matrix in our daily life

Bicky tiwari says

Sir…! I want to know what use of matrix in physical life.

vidhu Kumar Bansal says

Simply explained – Travel by local bus. You may have 15 destinations and 15 stops. Therefore what will be the fare from first stop to 15th stop or from third stop to tenth stop..

Thus a table of fare could be worked out using simple arithmetic matrix.

Tanveer alam says

In which side we don’t use matrix.

Sreya says

Really I want to know how matrices are used in daily life .how we will do that practically

john says

idk

kundan says

Matrix is very important role in our life

Nitesh Kumar Rai says

so helpfull for xii class and b.sc.

jageshwar ray says

Very helpful

..

Ani says

How is it applied in Petroleum engineering.pls i need specifics

phield says

Matrices can be used in a shop,were by goods are displayed in order of rows and columns

syed Junaid ul Hassan says

please tell me …..

how we can apply???

please clearfy with a example

anu sood says

it helped me in making time table of my institute and also maintaining accounts of students

Chris says

I would say the best use for matrices are by setting up experimental test sequences using orthogonal matrices, also know as Design of Expirements. If you have to setup a machine for instance for resistance welding and you want to establish the best parameters for optimizing multiple answers you can drastically reduce the ammount of tests that you need to run while at the same time getting good statistical evidence that the tests are valid.

For instance for resistance welding, I might decide on 3 variables to change. Current, Force, and Time. I then vary the current at two levels 3 kAmps and 5kAmps, force at 2 levels and 2 different times.

Instead of having to run 8 tests imperically, I can setup my test matrix and run 4 tests. With more variables and levels the time and cost savings increase. You also get a statistical understanding of how each variable actually changes the results helping to eliminate variables that dont really have an effect. When imperically testing you always have the question in the back of your mind. Are my results really valid? Did I do enough tests to rely on the result?

You can then optimize different the results for instance to get the strongest weld, the largest welding area, and for the lifetime of the welding electrodes.

I have sometime seen people trying to optimize certain processes where the desired results work counter to each other. With the Matrix data you can quickly see what is possible and what is not possible in a matter of hours instead of days, weeks, or months.

Mike DeHaan says

Thanks so much for your detailed note!

Guest says

Twitter and Facebook, as well as any social network are just huge matrices

divya hassani says

Cryptogram is awesome application of matrix

Md Matin says

Thankx… this is realy helpfull for my project..

prem says

Matrix mathematics applies to several branches of science, as well as different mathematical disciplines. Let’s start with computer graphics, then touch on science, and return to mathematics.

We see the results of matrix mathematics in every computer-generated image that has a reflection, or distortion effects such as light passing through rippling water.

Before computer graphics, the science of optics used matrix mathematics to account for reflection and for refraction.

Matrix arithmetic helps us calculate the electrical properties of a circuit, with voltage, amperage, resistance, etc.

In mathematics, one application of matrix notation supports graph theory. In an adjacency matrix, the integer values of each element indicates how many connections a particular node has.

The field of probability and statistics may use matrix representations. A probability vector lists the probabilities of different outcomes of one trial. A stochastic matrix is a square matrix whose rows are probability vectors. Computers run Markov simulations based on stochastic matrices in order to model events ranging from gambling through weather forecasting to quantum mechanics.

Matrix mathematics simplifies linear algebra, at least in providing a more compact way to deal with groups of equations in linear algebra.

Mike DeHaan says

Thank you for your insightful note.

Shouryodeep Chakraborty says

thank you , it has helped me in my project

ravindra kumar prajapati says

in the real life no real use of matrix if have then prove it .most respect………………

Ronald says

This helps me to calculate the number of points that a particular football team has, when goin to shop different locations you can compare prices this can be done by forming array of prices and then prices, do the multiplication.

mahesh says

nice

veronica ni ni win says

Thank you very much. It helps me for my papers.

Shouryodeep Chakraborty says

me too

kavya says

i find interesting solving matrices problems and therefore i use it in my school……..

BinaryStar34 says

I have no idea what this was supposed to be about and why it needed to be published.

For those who know and use linear algebra, this was an unnecessary article, and for those who don’t, it was a useless article, because they can not possibly understand how one can use matrices for circuit simulation.

That would require about one semester of university level math and one semester of university level circuit theory, just to explain the methods.

It’s even worse for group theory, 99.9% of which has absolutely nothing to do with matrices.

Guest says

shut up. this helped me loads.

Mike DeHaan says

Actually, I agree that there is a really tiny target area for introducing a math topic, where it’s interesting to those who don’t know the background, and/or useful for those who already work in the field.

I remember thinking that it would have been helpful if the original question were more specific. I tried to touch a variety of topics in a short article.

Sorry for the disappointment.

nath says

I follow this discussion. I want to know how can i use matrix to to make a forecast in a case of pollution of heavy metals in a river.

There are 8 variables measured for 3years. I need help