MATLAB: Augmented Matrices In this activity you will define an augmented matrix, find the number of pivot variables in the reduced system, and find the number of free variables in the solution to the linear system of equations.
Consider the linear system of equations:
2x + y = 3
x + 2y = 5
%Create the coefficient matrix C.
C = [2 1; 1 2]
%Create the column matrix d of constants. Remember, to create a column matrix, the rows are separated %by semicolons.
d = [3; 5]
%Create the augmented matrix [C I d]. Store this augmented matrix in Cd.
Cd = [Cd]
%Use the rref() command to reduce the augmented matrix. Store the reduced matrix in rowreducedCd, and %store the pivot variables in pivotvarsCd.
[rowreducedCd, pivotvarsCd] = rref(Cd)
%Warning: Look carefully at the reduced augmented matrix. If one of the pivot columns is the rightmost %column, the system of linear equation has no solution and no further analysis is possible.
%Do you run into any difficulties? Explain what is happening as a comment in your code.
%Use the size command to find the number of variables in the system of linear equations. Store this numbe %in numvars.
[numeqns, numvars] = size(C)
Store this number in numpivotvars.
%Use the size command to find the number of pivot variables.
[numrows, numpivotvars] = size(pivotvarsCd)
%Use subtraction to find the number of free variables in the solution to the system of linear equations.
%Store this number in numfreevars.
numfreevars = numvars - numpivotvars
Utilize the following linear system of equations for this activity.
X1 + 3x2 - 2x3 + 2xy = 0
2x1 + 6x2 - 5x3 - 2x4 + 4x3 - 3x6 = -1
x3 + 5x4 + 3x6 = 1
x1 + 3x2 + 4x4 + 2x5 + 9x6 = 3
Script
1 %Create the coefficient matrix A.
2
3 %Create the column matrix b of constants. Remember, to create a column matrix, the rows are separated
4 %by semicolons.
5
6 %Create the augmented matrix (A bl. Store this augmented matrix in Ab.
7
8 %Use the rref() command to reduce the augmented matrix. Store the reduced matrix in rowreducedAb, and
9 %store the pivot variables in pivotvarsAb.
10
11 %Warning: Look carefully at the reduced augmented matrix. If one of the pivot columns is the rightmos
12 %column, the system of linear equation has no solution and no further analysis is possible.
13
14 %Do you run into any difficulties? Explain what is happening as a comment in your code.
15
16 %Use the size command to find the number of variables in the system of linear equations. Store this nur
17 %in numvars.
18
19 %Use the size command to find the number of pivot variables. Store this number in numpivotvars.
20
21 %Use subtraction to find the number of free variables in the solution to the system of linear equations
22 %Store this number in numfreevars.