Given matrix
[1 2 3| -1]
[2 1 -1| 2]
[-1 0 5| 7]
What will be the new second row after the row operation: R2-2R1+R3?
a. [0, -3, -7 |4]
b. [-1, -3, -2 | 11]
c. [1, 1, 4 | 9]
d. [0, 0, 1 | 2]
R2-2R1 = [2 1 -1 | 2] - 2[1 2 3 | -1]
= [2 1 -1 | 2] - [2 4 6 | -2]
= [0 -3 -7 | 4]
Now do +R3
To find the new second row after the row operation R2-2R1+R3, we will perform the row operation on the given matrix.
The row operation R2-2R1+R3 means subtracting 2 times the first row from the second row and adding the third row.
Performing the row operation on the given matrix:
[1 2 3 | -1]
[2 1 -1 | 2]
[-1 0 5 | 7]
New second row:
2 - 2(1) + (-1) = 2 - 2 + (-1) = -1
1 - 2(2) + 0 = 1 - 4 + 0 = -3
-1 - 2(-1) + 5 = -1 + 2 + 5 = 6
7 - 2(-1) + 2(5) = 7 + 2 + 10 = 19
Therefore, the new second row after the row operation R2-2R1+R3 is [-1, -3, 6 | 19].
None of the options provided matches this answer, so none of the options are correct.
To perform the row operation R2 - 2R1 + R3 on the given matrix, follow these steps:
1. Start with the original matrix:
[1 2 3 | -1]
[2 1 -1 | 2]
[-1 0 5 | 7]
2. Multiply the first row, R1, by 2:
2 * R1 = [2 4 6 | -2]
3. Multiply the second row, R2, by 2:
-2 * R1 = [-4 -2 2 | 4]
4. Add the third row, R3, to the result of step 3:
[-4 -2 2 | 4]
+ [-1 0 5 | 7]
-----------------------------
[-5 -2 7 | 11]
5. Subtract the result of step 2 from the result of step 4:
[-5 -2 7 | 11]
- [2 4 6 | -2]
-----------------------------
[-7 -6 1 | 13]
The resulting matrix after performing the row operation is:
[-7 -6 1 | 13]
To determine the new second row, it is the second row of the resulting matrix. Therefore, the answer is option:
b. [-1, -3, -2 | 11]