se Gauss-Jordan elimination to solve the following linear system.x2 + x3 − 2x4 = −8x1 + 2x2 − x3 − x4 = 22x1 + 4x2 + x3 − 3x4 = −7x1 + 3x2 − 7x3 − x4 = 16
Question
se Gauss-Jordan elimination to solve the following linear system.x2 + x3 − 2x4 = −8x1 + 2x2 − x3 − x4 = 22x1 + 4x2 + x3 − 3x4 = −7x1 + 3x2 − 7x3 − x4 = 16
Solution
Sure, let's solve this system of linear equations using Gauss-Jordan elimination.
First, let's write the system of equations in augmented matrix form:
| 0 1 1 -2 | -8 |
| 1 2 -1 -1 | 2 |
| 2 4 1 -3 | -7 |
| 1 3 -7 -1 | 16 |
Next, we'll perform row operations to get this matrix in row-echelon form.
Step 1: Swap row 1 and row 2 to get x1 term at the top.
| 1 2 -1 -1 | 2 |
| 0 1 1 -2 | -8 |
| 2 4 1 -3 | -7 |
| 1 3 -7 -1 | 16 |
Step 2: Subtract 2*row1 from row3 and subtract row1 from row4.
| 1 2 -1 -1 | 2 |
| 0 1 1 -2 | -8 |
| 0 0 3 -1 | -11|
| 0 1 -6 0 | 14 |
Step 3: Swap row 2 and row 4.
| 1 2 -1 -1 | 2 |
| 0 1 -6 0 | 14 |
| 0 0 3 -1 | -11|
| 0 1 1 -2 | -8 |
Step 4: Subtract row2 from row4.
| 1 2 -1 -1 | 2 |
| 0 1 -6 0 | 14 |
| 0 0 3 -1 | -11|
| 0 0 7 -2 | -22|
Step 5: Divide row3 by 3 and row4 by 7.
| 1 2 -1 -1 | 2 |
| 0 1 -6 0 | 14 |
| 0 0 1 -1/3 | -11/3|
| 0 0 1 -2/7 | -22/7|
Step 6: Subtract row4 from row3.
| 1 2 -1 -1 | 2 |
| 0 1 -6 0 | 14 |
| 0 0 0 1/21 | 1/21|
| 0 0 1 -2/7 | -22/7|
Step 7: Multiply row3 by 21.
| 1 2 -1 -1 | 2 |
| 0 1 -6 0 | 14 |
| 0 0 0 1 | 1|
| 0 0 1 -2/7 | -22/7|
Step 8: Add 2/7*row3 to row4.
| 1 2 -1 -1 | 2 |
| 0 1 -6 0 | 14 |
| 0 0 0 1 | 1|
| 0 0 1 0 | -3|
Step 9: Subtract row4 from row1 and add 6*row4 to row2.
| 1 2 0 -1 | 5 |
| 0 1 0 0 | 2 |
| 0 0 0 1 | 1|
| 0 0 1 0 | -3|
Step 10: Add row4 to row1.
| 1 2 1 -1 | 2 |
| 0 1 0 0 | 2 |
| 0 0 0 1 | 1|
| 0 0 1 0 | -3|
Step 11: Subtract 2*row2 from row1.
| 1 0 1 -1 | -2 |
| 0 1 0 0 | 2 |
| 0 0 0 1 | 1|
| 0 0 1 0 | -3|
Step 12: Subtract row4 from row1.
| 1 0 0 -1 | 1 |
| 0 1 0 0 | 2 |
| 0 0 0 1 | 1|
| 0 0 1 0 | -3|
Step 13: Add row3 to row1.
| 1 0 0 0 | 2 |
| 0 1 0 0 | 2 |
| 0 0 0 1 | 1|
| 0 0 1 0 | -3|
So, the solution to the system of equations is x1 = 2, x2 = 2, x3 = -3, x4 = 1.
Similar Questions
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