HW11P4 (15 points) - Linear System Consistency Consider the linear system: x1 2x
ID: 3138668 • Letter: H
Question
HW11P4 (15 points) - Linear System Consistency Consider the linear system: x1 2x2 + 4x3 x4 11 x 2x2 +3x4-1 -X1 + 3x2 + X3-2x4 =-2 2x1 + 2x2 - 2x38 a) (2 pts) Rewrite the system of equations in Ax b form. b) (6 pts) By hand, find rank(A) and rank(Alb) by using Gaussian elimination to reduce A and (Alb) to row echelon form (ref). Check your answer using MATLAB. Write down the MATLAB commands you used to determine your answer. (3 pts) Determine whether the system of equations is consistent or inconsistent. If the system is consistent, determine, without solving the linear system, whether it has a unique solution or infinitely many solutions. (4 pts) Solve the linear system (i.e. determine x1, x2, X3, and x4) by performing back substitution. c) d)Explanation / Answer
(a). The given linear system can be written in matrix form as AX = b, where A =
1
2
4
1
-1
2
0
3
-1
3
1
-2
-2
2
-2
0
X = (x1,x2,x3,x4)T, and b = (11,1,-2,-8)T.
(b). We can reduce the matrix M = [A|b] to its RREF as under:
Add 1 times the 1st row to the 2nd row
Add 1 times the 1st row to the 3rd row
Add 2 times the 1st row to the 4th row
Multiply the 2nd row by ¼
Add -5 times the 2nd row to the 3rd row
Add -6 times the 2nd row to the 4th row
Multiply the 3rd row by -1/6
Add 4 times the 3rd row to the 4th row
Add -1 times the 3rd row to the 2nd row
Add -1 times the 3rd row to the 1st row
Add -2 times the 2nd row to the 1st row
Then the RREF of M = [A|b] is
1
0
2
0
6
0
1
1
0
2
0
0
0
1
1
0
0
0
0
0
The RREF of A is
1
0
2
0
0
1
1
0
0
0
0
1
0
0
0
0
(c ).It is apparent from the RREF of M = [A|b]that the given linear system is consistent and that it has infinite solutions.
(d). The given linear system is equivalent to x1+2x3= 6 or, x1 = 6-2x3 , x+x3 = 2 or, x2 = 2-x3 and x4 = 0. Then, X = (x1,x2,x,x4)T=(6-2x3,2-x3,x3,0)T = (6,2,0,0)T+t(-2,-1,1,0)T where t is an arbitrary real number.
1
2
4
1
-1
2
0
3
-1
3
1
-2
-2
2
-2
0
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