Use Of Matrices In Finding Area Of Triangle
We can use matrices to find the area of any triangle where the vertices of the triangle have been given.
Lets suppose that we have a triangle ABC with vertices A , B , C
Now the area of the triangle ABC ,
Area of triangle ABC can be given by the determinant= \ |
Use of Matrices for Collinear Point
Matrices can be used to check where any three given points are collinear or not. Three points suppose A , B , C are collinear if they do not form a triangle, that is the area of the triangle should be equal to zero.
The points A,B,C are collinear if \ vanishes. |
1. What are the applications of matrices?
They are used for plotting graphs, statistics and also to do scientific studies and research in almost different fields. Matrices can also be used to represent real world data like the population of people, infant mortality rate, etc. They are the best representation methods for plotting surveys.
2. What is the application of matrices in engineering?
Application of matrices in Engineering
Transformation matrices are commonly used in computer graphics and image processing. Matrices are used in computer generated images that have a reflection and distortion effect such as high passing through ripping water. This is how Application of matrices in engineering is used.
3. What is the application of matrices in business and economics?
Upper And Lower Triangular Matrices
The diagonal of a matrix always refers to the leading diagonal. The leading diagonal in a matrix helps to define two other types of matrix: lower-triangular matrices and upper triangular matrices. A lower-triangular matrix has numbers beneath the diagonal; an upper-triangular matrix has numbers above the diagonal.A diagonal matrix is both a lower-diagonal and a lower-diagonal matrix.
How Do You Do Matrix Math
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Using The Determinant To Get The Area Of A Triangle
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.
Lets say we have the three coordinate points of that triangle, \,\left\), and \\). The formula for the area of the triangle bounded by those points is:
\,\,\left\,\,\text\,\left=\pm \frac\left| } _}} & _}} & 1 \\ _}} & _}} & 1 \\ _}} & _}} & 1 \end} \right|\)
Lets do an example:
Find the area of the triangle bounded by the points , \,\left\) and \\).; ; ; ; ; ;
Solution:
Use the equation above to find the area:
\=\frac\left=14.5\end\)
We take the positive only since the determinant is positive.
Matrix Word Problem When Tables Are Not Given:
Sometimes youll get a matrix word problem where just numbers are given; these are pretty tricky. Here is one:
An outbreak of Chicken Pox hit the local public schools. Approximately 15% of the male and female juniors and 25% of the male and female seniors are currently healthy, 35% of the male and female juniors and 30% of the male and female seniors are currently sick, and 50% of the male and female juniors and 45% of the male and female seniors are carriers of Chicken Pox.;
There are 100 male juniors, 80 male seniors, 120 female juniors, and 100 female seniors.
Using two matrices and one matrix equation, find out how many males and how many females are healthy, sick, and carriers.
Solution:
The best way to approach these types of problems is to set up a few manual calculations and see what were doing. For example, to find out how many healthy males we would have, wed set up the following equation and do the calculation: \+.25=35\). Likewise, to find out how many females are carriers, we can calculate:; \+.45=105\).
We can come up with the following matrix multiplication:
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!
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Use Of Matrices In Cryptography
Cryptography is the technique to encrypt data so that only the relevant person can get the data and relate information. In earlier days, video signals were not used to encrypt. Anyone with satellite dish was able to watch videos which results in the loss for satellite owners, so they started encrypting the video signals so that only those who have video ciphers can unencrypt the signals. This encryption is done by using an invertible key that is not invertible then the encrypted signals cannot be unencrypted and they cannot get back to their original form. This process is done using matrices. A digital audio or video signal is firstly taken as a sequence of numbers representing the variation over time of air pressure of an acoustic audio signal. The filtering techniques are used which depends on matrix multiplication.;
A Florist Must Make 5 Identical Bridesmaid Bouquets Systems Problem
Heres 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. She has $610 to spend and wants 24 flowers for each bouquet. Roses cost $6 each, tulips cost $4 each, and lilies cost $3 each. 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?
Solution:
Lets look at the question that is being asked and define our variables:; Let \;the number of roses, \;the number of tulips, and \;the number of lilies. Lets put the money terms together, and also the counting terms together:
\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text\\r+t+l=5\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text\end\)
Now lets put the system in matrices and on the calculator:
For all the bouquets, well have 80 roses, 10 tulips, and 30 lilies.
For one bouquet, well have;\;of the flowers, so well have 16 roses, 2 tulips, and 6 lilies.
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What Is Adjoint Of A Square Matrix
In linear algebra, the adjugate or classical adjoint of a square matrix is the transpose of its cofactor matrix. The adjugate has sometimes been called the adjoint, but today the adjoint of a matrix normally refers to its corresponding adjoint operator, which is its conjugate transpose.
Adding And Subtracting Matrices
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:
\displaystyle A+B=B+A
\displaystyle +C=A+
Adding matrices is very simple. Just add each element in the first matrix to the corresponding element in the second matrix.
\displaystyle \begin 1 & 2 & 3 \\ 4 & 5 & 6 \end+\begin 10 & 20 & 30 \\ 40 & 50 & 60 \end=\begin 11 & 22 & 33 \\ 44 & 55 & 66 \end
Note that element ;in the first matrix, 1, adds to element x_;in the second matrix, 10, to produce element x_;in the resultant matrix, 11. Also note that both matrices being added are 2\times 3, and the resulting matrix is also 2\times 3. You cannot add two matrices that have different dimensions.
As you might guess, subtracting works much the same way except that you subtract instead of adding.
\displaystyle \begin 10 & -20 & 30 \\ 40 & 50 & 60 \end-\begin 1 & -2 & 3 \\ 4 & -5 & 6 \end=\begin 9 & -18 & 27 \\ 36 & 55 & 54 \end
Once again, note that the resulting matrix has the same dimensions as the originals, and that you cannot subtract two matrices that have different dimensions. Be careful when subtracting with signed numbers.
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X 2 Matrix Multiplication
So what was the point of learning the dot product? Well, we will be using the dot product when we multiply two matrices together. When multiplying a matrix with another matrix, we want to treat rows and columns as a vector. More specifically, we want to treat each row in the first matrix as vectors, and each column in the second matrix as vectors. Let’s do an example.
Question 3: Find
2 c_, c_ c1,c2 are first and second columns. Now we are going to treat each row and column we see here as a vector.
Notice here that multiplying a 2 x 2 matrix with another 2 x 2 matrix gives a 2 x 2 matrix. In other words, the matrix we get should have 4 entries.
How do we exactly get the first entry? Well, notice that the first entry is located on the first row and first column. So we simply take the dot product of r
c2. This gives us:
If we are to keep locating all the entries and doing the dot product corresponding to the rows and columns, then we get the final result.
We are done! Notice that the bigger the matrices are, the more tedious matrix multiplication becomes. This is because we have to deal with more and more numbers! In general, the matrix multiplication formula for 3 x 3 matrices is
How Do I Properly Do Math With Matrices For Opengl Transformations
I’m really confused how matrices work and how to use them possible. Just as a test, I am trying to make a rectangle rotate anti-clockwise while it is moving towards a corner, also the rectangle is half the size. Basically everything doesnt work expect rotation.
If you are wondering, I am not using GLM and i don’t want to use GLM. I feel like I need to try myself but now im seriously stuck. Ive tried rearranging how I do the transformations, but I get random results.
Here is the code in main.cpp it is a custom game engine so here is just the relevant part
Matrix4 transform = Matrix4; // the matrixfloat d = 5, g = 1;void OnUserUpdate override
Here is the full matrix class. Vector3 is just float x, y, z. There is only a constructor.
template <typename Number> float ToRadians struct Matrix4 public:float m;Matrix4 Matrix4 void Set }void Projection void Translate void Rotate void Scale };
As I said I am trying to make a rectangle rotate anti-clockwise while it is moving towards a corner, and the rectangle is half the size. I used a few sources, but honestly I didn’t fully understand everything
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Schools Wikipedia Selection Related Subjects: Mathematics
In mathematics, a matrix is a rectangular table of numbers or, more generally, a table consisting of abstract quantities that can be added and multiplied. Matrices are used to describe linear equations, keep track of the coefficients of linear transformations and to record data that depend on two parameters. Matrices can be added, multiplied, and decomposed in various ways, making them a key concept in linear algebra and matrix theory.
In this article, the entries of a matrix are real or complex numbers unless otherwise noted.
Properties Of Transpose Matrices
Properties for transpose matrices are similar to the basic number properties that you encountered in basic algebra . The basic properties for matrices are:
- T = A: the transpose of a transpose matrix is the original matrix.
- T = AT + BT: The transpose of two matrices added together is the same as the transpose of each individual matrix added together.
- T = rAT: when a matrix is multiplied by a scalar element, it doesnt matter which order you transpose in .
- T = BT AT: the transpose of two matrices multiplied together is the same as the product of their transpose matrices in reverse order.
- T = -1: the transpose and the inverse of a matrix can be performed in any order.
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Use Of Matrices In Computer Graphics
Earlier architecture, cartoons, automation were done by hand drawings but nowadays they are done by using computer graphics. Square matrices very easily represent linear transformation of objects. They are used to project three dimensional images into two dimensional planes in the field of graphics. In Graphics, digital image is treated as a matrix to start with. The rows and columns of the matrix correspond to rows and columns of pixels and the numerical entries correspond to the pixels color values. Using matrices to manipulate a point is a common mathematical approach in video game graphics Matrices are also used to express graphs. Every graph can be represented as a matrix, each column and each row of a matrix is a node and the value of their intersection is the strength of the connection between them. Matrix operations such as translation, rotation and sealing are used in graphics. For transformation of a point we use the equation
The Additive Identity Matrix
When people talk about the Identity Matrix they are usually talking about the multiplicative identity matrix. However, there is another type: the additive identity matrix. When this matrix is added to another, you end up the original matrix. Not surprisingly, every element in these matrices are zeros. Therefore, they are sometimes called the zero matrix.
An additive identity matrix for a 3 * 3 matrix.
For an overview of finding inverses, watch this short video:
Inverse matrices are the same idea as reciprocals. In elementary algebra , you came across the idea of a reciprocal: one number multiplied by another can equal 1.
Image courtesy of LTU
Identity Matrix
Step 1: Find the adjugate of the matrix. The adjugate of the matrix can be found by rearranging one diagonal and taking negatives of the other:
To find the adjugate of a 2×2 matrix, swap diagonals a and d, then swap the signs of c and d.
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Matrix Algebra: Addition And Subtraction
The size of a matrix is also called the matrix dimension or matrix order. If you want to add two matrices, their dimensions must be exactly the same. In other words, you can add a 2 x 2 matrix to another 2 x 2 matrix but not a 2 x 3 matrix. Adding matrices is very similar just regular addition: you just add the same numbers in the same location .
A note on notation: A worksheet uses column letters and row numbers to give a cell location like A1 or D2. Its typical for matrices to use notation like gij which means the ith row and jth column of matrix G.
Matrix subtraction works exactly the same way.
What Is An Identity Matrix
An identity matrix is a square matrix with 1s as the elements in the main diagonal from top left to bottom right and zeros in the other spaces. When you multiply a square matrix by an identity matrix, it leaves the original square matrix unchanged. For example:
The idea is similar to the identity element. In basic math, the identity element leaves a number unchanged. For example, in addition the identity element is 0, because 1 + 0 = 1, 2 + 0 = 2 etc. and in multiplication, the identity element is 1 because any number multiplied by 1 equals that number . In more formal terms, if x is a real number, then the number 1 is called the multiplicative identity because 1 * x = x and x * 1 = x. By the same logic, the identity matrix I gets its name because, for all matrices A, I * A = A and A * I = A.
In matrix algebra, the identity element is different depending on the size of the matrix you are operating on; unlike the singular 1 for the multiplicative identity and 0 for additive identity, there is no single identity matrix for all matrices. For any n * n matrix there is an identity matrix In * n. The main diagonal will always have 1s and the remaining spaces will all be zeros. The following image shows identity matrices for a 2 x 2 matrix and a 5 x 5 matrix:
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Relationship To Linear Maps
Linear maps Rn Rm are equivalent to m-by-n matrices, as described above. More generally, any linear map f: V W between finite-dimensionalvector spaces can be described by a matrix A = , after choosing basesv1, …, vn of V, and w1, …, wm of W , which is such that
- f n . _)=\sum _^a_\mathbf _\qquad }\ j=1,\ldots ,n.}
In other words, column j of A expresses the image of vj in terms of the basis vectors wi of W; thus this relation uniquely determines the entries of the matrix A. The matrix depends on the choice of the bases: different choices of bases give rise to different, but equivalent matrices. Many of the above concrete notions can be reinterpreted in this light, for example, the transpose matrix AT describes the transpose of the linear map given by A, with respect to the dual bases.
These properties can be restated more naturally: the category of all matrices with entries in a field k with multiplication as composition is equivalent to the category of finite-dimensional vector spaces and linear maps over this field.
More generally, the set of m×n matrices can be used to represent the R-linear maps between the free modules Rm and Rn for an arbitrary ring R with unity. When n;=;m composition of these maps is possible, and this gives rise to the matrix ring of n×n matrices representing the endomorphism ring of Rn.
- MTM = I,