In: Math
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
.
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