Answer: Therefore, the reduced row echelon form for the matrix shown is:
[0 1 0 0 3]
[0 0 0 0 0]
[0 0 0 0 0]
[0 0 0 0 0]
Explanation:
To determine the reduced row echelon form of the given matrix, we need to perform row operations to simplify the matrix as much as possible. Let's denote the matrix as:
[a b c d e]
Using the given options (A, B, C, D), we can test each one and see which option results in the reduced row echelon form.
Option A:
3 4 0 -5 11
- Perform row operations to simplify the matrix:
- R1 → R1/3 (divide the first row by 3)
- R2 → R2 - 4R1 (subtract 4 times the first row from the second row)
- R3 → R3 + 5R1 (add 5 times the first row to the third row)
- R4 → R4 + 11R1 (add 11 times the first row to the fourth row)
After performing these row operations, the matrix becomes:
[1 4/3 0 -5/3 11/3]
[0 -4 -1 5 -1]
[0 -5 3 0 8]
[0 -19 11 0 22]
Option B:
1 0 0 2 5
- Perform row operations to simplify the matrix:
- R1 → R1 - 3R2 (subtract 3 times the second row from the first row)
- R2 → R2 + 4R1 (add 4 times the first row to the second row)
- R3 and R4 remain unchanged
After performing these row operations, the matrix becomes:
[1 0 0 2 5]
[0 4 0 -2 -3]
[0 0 0 0 0]
[0 0 0 0 0]
Option C:
0 1 0 0 3
- Perform row operations to simplify the matrix:
- R1 and R2 remain unchanged
- R3 → R3 - 5R2 (subtract 5 times the second row from the third row)
- R4 → R4 + 11R2 (add 11 times the second row to the fourth row)
After performing these row operations, the matrix becomes:
[0 1 0 0 3]
[0 0 0 0 0]
[0 0 0 0 0]
[0 0 0 0 0]
Option D:
0 0 1 0 7
- Perform row operations to simplify the matrix:
- R1, R2, and R3 remain unchanged
- R4 → R4 - 11R3 (subtract 11 times the third row from the fourth row)
After performing these row operations, the matrix becomes:
[0 0 1 0 7]
[0 0 0 0 0]
[0 0 0 0 0]
[0 0 0 0 0]
Based on the row operations performed, we can conclude that option C (a = 0, b = 1, c = 0, d = 0, e = 3) is the correct reduced row echelon form of the given matrix.