Matrix solutions of the linear algebraic equation, MATLAB in Engineering

Matrix solutions to systems of the linear algebraic equations:

The linear algebraic equation is an equation of the form

a1x1 + a2x2 + a3x3   .  .  .  .  anxn = b

Where a's are the constant coefficients, the x's are the unknowns, and b be constant. A solution is a series of numbers s1, s2, and s3 which satisfy the equation. The illustration is as follows,

4x1 +  5x2 - 2x3 = 16

is such an equation in which there are 3 unknowns: x1, x2, and x3. The One solution to this equation is x1 = 3, x2 = 4, and x3 = 8, as 4 * 3 + 5 * 4 - 2 * 8 is equal to 16.

The system of linear algebraic equations is a set of equations of the form:

621_Matrix solutions of the linear algebraic equation.png

 

This is known as m × n system of equations; there are m equations and n unknowns.

As of the way that matrix multiplication works, such equations can be presented in matrix form as Ax = b here A is a matrix of the coefficients, x is the column vector of the unknowns, and b is the column vector of constants from the right-hand side of the equations:

A solution set is a set of all the possible solutions to the system of equations (all sets of values for the unknowns which solve the equations). All the systems of linear equations have either:

  •  No solutions
  •  One solution
  •  Infinitely many solutions

The one of the main concepts of the subject of linear algebra is the various techniques of solving (or trying to solve!) systems of the linear algebraic equations. The MATLAB has many functions which assist in this process.

The system of equations has been once written in matrix form, what we want is to evaluate the equation Ax = b for the unknown x. To do this, we require to isolate x on one side of the equation. If we were working with scalars, then we divide both sides of the equation by x. However, with the MATLAB we can use the divided into operator to do this. Though, most languages cannot do this with matrices, therefore we rather multiply both sides of the equation by the inverse of the coefficient matrix A:

A-1 A x = A-1 b

Then, as multiplying a matrix by its inverse results in the identity matrix I, and since multiplying any matrix by I answers in the original matrix, we contain:

I x = A-1 b

or

x = A-1 b

This means that the column vector of unknown x is found as the inverse of matrix A multiplied by the column vector b. Therefore, if we can find the inverse of A, we can resolve for the unknown in x.

Posted Date: 10/22/2012 2:45:36 AM | Location : United States







Related Discussions:- Matrix solutions of the linear algebraic equation, Assignment Help, Ask Question on Matrix solutions of the linear algebraic equation, Get Answer, Expert's Help, Matrix solutions of the linear algebraic equation Discussions

Write discussion on Matrix solutions of the linear algebraic equation
Your posts are moderated
Related Questions
Plotting File data: It is frequently essential to read data from a file and plot it. Generally, this entails knowing the format of the file. For illustration, let us suppose t

Algorithm for appex subfunction: The algorithm for appex subfunction is as shown:  Receives x & n as the input arguments.  Initializes a variable for running sum of t

Storing Strings in Cell Arrays: The one good application of a cell array is to store strings of various lengths. As cell arrays can store various types of values in the elemen

. Generate the following signal, x(n)=1+cos((25*pi*n)/100),0 Compute the DTFT of x[n] for w=0:0.01:2*pi Plot the Real part, imaginary part, the amplitude and phas

ischar function: The ischar function return the logical true if an array is a character array, or logical false if not. >> vec = 'EK127'; >> ischar(vec) ans =

Defined a variable in work space: The variables defined in the script will become a part of the workspace: >> clear >> who >> mysummfile    15 >> who

Image Processing: The Images are represented as grids, or matrices, of picture elements (known as pixels). In MATLAB an image usually is represented as a matrix in which each

Displaying expressions: The good-looking function will show such expressions by using exponents; for illustration, >> b = sym('x^2') b = x^2 >> pretty(b)


Creating the structure Variables: Creating a structure variable can be accomplished by simply storing the values in fields by using assignment statements, or by using the stru