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Express the following systems of equations in matrix form as A x = b. What is...

Express the following systems of equations in matrix form as A x = b. What is the rank of A in each case (hint: You can find the rank of a matrix A by using rank = qr(A)$rank in R. For each problem, find the solution set for x using R.

a. X1+5X2+2X3=5

4X1-X2+3X3=-8

6X1-2X2+X3=0

b. 3X1-5X2+6X3+X4=7

4X1+2X3-3X4=5

X2-3X3+7X4=0

X2+3X4=5

Solutions

Expert Solution

.

system is

augmented matrix is

1 5 2 5
4 -1 3 -8
6 -2 1 0

convert into Reduced Row Eschelon Form...

Add (-4 * row1) to row2

1 5 2 5
0 -21 -5 -28
6 -2 1 0


Add (-6 * row1) to row3

1 5 2 5
0 -21 -5 -28
0 -32 -11 -30


Divide row2 by -21

1 5 2 5
0 1 5/21 4/3
0 -32 -11 -30


Add (32 * row2) to row3

1 5 2 5
0 1 5/21 4/3
0 0 -71/21 38/3


Divide row3 by -71/21

1 5 2 5
0 1 5/21 4/3
0 0 1 -266/71


Add (-5/21 * row3) to row2

1 5 2 5
0 1 0 158/71
0 0 1 -266/71


Add (-2 * row3) to row1

1 5 0 887/71
0 1 0 158/71
0 0 1 -266/71


Add (-5 * row2) to row1

1 0 0 97/71
0 1 0 158/71
0 0 1 -266/71

for every 3 column of matrix there is a pivot entry so rank of matrix is 3

and solution is

.

.

system is

augmented matrix is

3 -5 6 1 7
4 0 2 -3 5
0 1 -3 7 0
0 1 0 3 5

convert into  Reduced Row Eschelon Form...

Divide row1 by 3

1 -5/3 2 1/3 7/3
4 0 2 -3 5
0 1 -3 7 0
0 1 0 3 5


Add (-4 * row1) to row2

1 -5/3 2 1/3 7/3
0 20/3 -6 -13/3 -13/3
0 1 -3 7 0
0 1 0 3 5


Divide row2 by 20/3

1 -5/3 2 1/3 7/3
0 1 -9/10 -13/20 -13/20
0 1 -3 7 0
0 1 0 3 5


Add (-1 * row2) to row3

1 -5/3 2 1/3 7/3
0 1 -9/10 -13/20 -13/20
0 0 -21/10 153/20 13/20
0 1 0 3 5


Add (-1 * row2) to row4

1 -5/3 2 1/3 7/3
0 1 -9/10 -13/20 -13/20
0 0 -21/10 153/20 13/20
0 0 9/10 73/20 113/20


Divide row3 by -21/10

1 -5/3 2 1/3 7/3
0 1 -9/10 -13/20 -13/20
0 0 1 -51/14 -13/42
0 0 9/10 73/20 113/20


Add (-9/10 * row3) to row4

1 -5/3 2 1/3 7/3
0 1 -9/10 -13/20 -13/20
0 0 1 -51/14 -13/42
0 0 0 97/14 83/14


Divide row4 by 97/14

1 -5/3 2 1/3 7/3
0 1 -9/10 -13/20 -13/20
0 0 1 -51/14 -13/42
0 0 0 1 83/97


Add (51/14 * row4) to row3

1 -5/3 2 1/3 7/3
0 1 -9/10 -13/20 -13/20
0 0 1 0 817/291
0 0 0 1 83/97


Add (13/20 * row4) to row2

1 -5/3 2 1/3 7/3
0 1 -9/10 0 -91/970
0 0 1 0 817/291
0 0 0 1 83/97


Add (-1/3 * row4) to row1

1 -5/3 2 0 596/291
0 1 -9/10 0 -91/970
0 0 1 0 817/291
0 0 0 1 83/97


Add (9/10 * row3) to row2

1 -5/3 2 0 596/291
0 1 0 0 236/97
0 0 1 0 817/291
0 0 0 1 83/97


Add (-2 * row3) to row1

1 -5/3 0 0 -346/97
0 1 0 0 236/97
0 0 1 0 817/291
0 0 0 1 83/97


Add (5/3 * row2) to row1

1 0 0 0 142/291
0 1 0 0 236/97
0 0 1 0 817/291
0 0 0 1 83/97

for every 4 column of matrix there is a pivot entry so rank of matrix is 4

and solution is


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