In: Math
Solve the linear system of equations Ax = b, and give the rank of the matrix A, where A =
1 | 1 | 1 | -1 |
0 | 1 | 0 | 2 |
1 | 0 | 1 | 1 |
b =
1 |
2 |
3 |
augmented matrix is
1 | 1 | 1 | -1 | 1 |
0 | 1 | 0 | 2 | 2 |
1 | 0 | 1 | 1 | 3 |
convert into Reduced Row Eschelon Form...
Add (-1 * row1) to row3
1 | 1 | 1 | -1 | 1 |
0 | 1 | 0 | 2 | 2 |
0 | -1 | 0 | 2 | 2 |
Add (1 * row2) to row3
1 | 1 | 1 | -1 | 1 |
0 | 1 | 0 | 2 | 2 |
0 | 0 | 0 | 4 | 4 |
Divide row3 by 4
1 | 1 | 1 | -1 | 1 |
0 | 1 | 0 | 2 | 2 |
0 | 0 | 0 | 1 | 1 |
Add (-2 * row3) to row2
1 | 1 | 1 | -1 | 1 |
0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 1 |
Add (1 * row3) to row1
1 | 1 | 1 | 0 | 2 |
0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 1 |
Add (-1 * row2) to row1
1 | 0 | 1 | 0 | 2 |
0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 1 |
reduced system is
there are 3 pivot entry at first second and fourth column
hence rank is 3
.
....................free
general solution is