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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

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

milcah answered 3 years ago

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