. (a) What are the different methods to solve a system of linear equations? Brie

Question

. (a) What are the different methods to solve a system of linear equations? Briefly describe the steps needed in each method.

(b) What is a linear programming problem with multiple optimal solutions? How do we find if a given linear programming problem has multiple optimal solutions? Give a real world example of a linear programming problem where multiple optimal solutions may occur.

(c) What are the different parts of a linear programming problem? Briefly describe each part. Given a real world problem, how will you formulate each part to develop a linear programming model for it?

(d) Briefly describe the important parts of each step needed to make a decision using decision sciences models.

Explanation / Answer

There are several models to solve a system of linear equations. Here are some of them:

Let us discuss briefly, the steps involved in these methods.

Gauss Elimination Method:

We write the given system of linear equations in matrix form Ax=b

Now, we write a new matrix by [A:b]

Now, we perform row operations on this matrix to convert matrix A to an identity matrix of same order.

Once we have obtained the identity matrix from row operations, the resultant matrix b (that is last column) will be our desired solution.

LU Decomposition Method:

In this method, we write the system in matrix form Ax=b

Then we decompose the matrix A in its LU decomposition form so that we have A=LU

We write the system as LUx=b

Now, we consider Ux=y

Therefore, we get two very easily solvable systems, respectively, Ly=b and Ux=y.

Using Ly=b, we first solve for y. Then plugging that value of y in Ux=y, we get the value of x. That is our final solution of the system.

Cramer’s Rule:

In cramar’s rule, we write the system as Ax=b

Then we find determinants of Matrix A as well as the other matrices found by replaced each column of A by ‘b’ one after another. Finally, we divide those determinants by determinant of matrix A to get the corresponding solutions.

In a nutshell we can write the solution of cramer’s rule as:

Xi = Det(Ai)/Det(A)

Matrix Inversion Method:

In matrix inversion method, we write the system of linear equations as Ax=b

Then we find the inverse matrix of matrix A.

Then we multiply the equation Ax=b by A-1

We get,

A-1Ax=A-1b

Ix=A-1b

X = A-1b

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