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Transcript of Matrices
-In mathematics, a matrix (plural matrices) is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. The individual items in a matrix are called its elements or entries.
An augmented matrix for a system of equations is a matrix of numbers in which each row represents the constants from one equation and each column represents all the coefficients for a single variable.
Reduced Row Echelon Form
It's a method to solve augmented matrices.
A matrix is in Reduced Row Echelon Form if:
- It is in row echelon form.
-Every leading coefficient (pivot entry) is 1 and is the only nonzero entry in its column.
Unlike the row echelon form (that can be put in an (equivalent) echelon form by adding a scalar multiple of a row to one of the above rows) the reduced row echelon form of a matrix is unique.
When coding and decoding matrices, you'll be using three matrices in the process:
- A Message Matrix
-An Encoding Matrix
- The Enconded Matrix
The sum A+B of two m-by-n matrices A and B is calculated entrywise:
(A + B)i,j = Ai,j + Bi,j, where 1 ≤ i ≤ m and 1 ≤ j ≤ n.
The two matrices must be the same size, i.e. the rows must match in size, and the columns must match in size.
Negative and Subtracting
Subtracting or addition of a negative matrix:
Scalar multiplication or Multiply by a Constant
The scalar multiplication cA of a matrix A and a number c (also called a scalar in the parlance of abstract algebra) is given by multiplying every entry of A by c:
(cA)i,j = c · Ai,j.
The transpose of an m-by-n matrix A is the n-by-m matrix AT (also denoted Atr or tA) formed by turning rows into columns and vice versa:
Multiplying by Another Matrix
To multiply a matrix by another matrix we need to do the "dot product" of rows and columns .
Multiplication only happens if the number of columns of the left matrix is the same as the number of rows of the right matrix.
The "Dot Product" is where we multiply matching members, then sum up:
(1, 2, 3) • (7, 9, 11) = 1×7 + 2×9 + 3×11 = 58
We match the 1st members (1 and 7), multiply them, likewise for the 2nd members (2 and 9) and the 3rd members (3 and 11), and finally sum them up.
To multiply an m×n matrix by an n×p matrix, the ns must be the same, and the result is an m×p matrix.
Note: matrix multiplication is not commutative, therefore:
Well we don't actually divide matrices, we do it this way:
A/B = A × (1/B) = A × B^-1
where B^-1 means the "inverse" of B.
So we don't divide, instead we multiply by an inverse.
Inverse of a Matrix
The Inverse of a Matrix is the same idea as the reciprocal of a number:
But we don't write 1/A (because we don't divide by a Matrix!), instead we write A-1 for the inverse:
When you multiply a Matrix by its Inverse you get the Identity Matrix
A x×A^-1 = I
The "Identity Matrix" is the matrix equivalent of the number "1"
It is "square" (has same number of rows as columns),
It has 1s on the diagonal and 0s everywhere else.
It's symbol is the capital letter I.
Calculating the Inverse
Swap the positions of a and d, put negatives in front of b and c, and divide everything by the determinant (ad-bc)
The Inverse May Not Exist:
When the determinant of a matrix is 0 the matrix has no inverse.
Such a Matrix is called "Singular", which only happens when the determinant is zero.
It happens when the matrix does not add new information.
Solving and augmented matrix:
Write a message and codify it
- I am a rock star
Write the "Number sentence" as your "Message Matrix"
- Always try to use 2x2, 3x3, 4x4 matrices
Chose an encoding matrix to multiply by your message matrix. (MUST BE A SQUARE MATRIX)
Write the " Encoded Matrix" as a string
Decoding the Matrix:
- The inverse of the Encoding Matrix
The inverse of the Encoding Matrix:
Transform the string of numbers into a matrix. Since the inverse of the encoded matrix is a 3x3 matrix, the string matrix will have 3 columns.
Multiply the "The String Matrix" by the "Inverse of the Encoding Matrix"
Write the Result as a string message and plug into the Character Key:
Equal to 0
I am a rock star.
Note: This does not always mean that the left of the matrix will be an identity matrix, as this example shows.
Row echelon Form
Real Life Applications
Inverse of a Matrix:
A group took a trip on a bus, at $3 per child and $3,20 per adult for a total of $118,40.
They took the train back at $3,50 per child and $3,60 per adult for a total of $135,20. How many children, and how many adults?
So to solve it we need the inverse of "A":
Now that we have the inverse:
A: 16 children and 22 adults
There are numerous applications of matrices, both in mathematics and other sciences.
In game theory and economics, the payoff matrix encodes the payoff for two players, depending on which out of a given (finite) set of alternatives the players choose.
Complex numbers can be represented by particular real 2-by-2 matrices
Computer graphics uses matrices both to represent objects and to calculate transformations of objects using affine rotation matrices to accomplish tasks such as projecting a three-dimensional object onto a two-dimensional screen, corresponding to a theoretical camera observation.
Chemistry makes use of matrices in various ways, particularly since the use of quantum theory to discuss molecular bonding and spectroscopy.
A general application of matrices in physics is to the description of linearly coupled harmonic systems. The equations of motion of such systems can be described in matrix form, with a mass matrix multiplying a generalized velocity to give the kinetic term, and a force matrix multiplying a displacement vector to characterize the interactions.
Geometrical optics provides further matrix applications. In this approximative theory, the wave nature of light is neglected. The result is a model in which light rays are indeed geometrical rays. If the deflection of light rays by optical elements is small, the action of a lens or reflective element on a given light ray can be expressed as multiplication of a two-component vector with a two-by-two matrix called ray transfer matrix: the vector's components are the light ray's slope and its distance from the optical axis, while the matrix encodes the properties of the optical element. Actually, there are two kinds of matrices, a refraction matrix describing the refraction at a lens surface, and a translation matrix, describing the translation of the plane of reference to the next refracting surface, where another refraction matrix applies. The optical system, consisting of a combination of lenses and/or reflective elements, is simply described by the matrix resulting from the product of the components' matrices
I could conclude that:
- Matrices are useful in solving systems of linear equations and problems with different variables and scenarios.
- It's easy to understand, therefore when solving by matrices you need to be careful with the algebra and the calculations ( you can make easy mistakes)
-It appears for the first time between 300 BC and AD 200, Nine Chapters of the Mathematical Art (Chiu Chang Suan Shu).
- How they came up with the operations related to matrices, such as : inverted matrices
Beatriz Zancanela 10th
Matrix Row Operations
Switching Rows: You can switch the rows of a matrix to get a new matrix.
Multiplying a Row by a Number: You can multiply any row by a number. (This means multiplying every entry in the row by the same number.)
Adding Rows: add two rows together, and replace a row with the result.
To find the inverse of a 3x3 matrix play around with the rows until we make Matrix A into the Identity Matrix I. And by also doing the changes to an Identity Matrix.
Reduced Echelon Form:
limited variables solution
Reduced Echelon Form:
unlimited variables solutions
Reduced Echelon Form:
no variable solution