The reduced row echelon form of the matrix
-1/2 1/2 3/2
-3/2 2 3/2
1/2 -3/2 9/2
is?
To find the reduced row echelon form of a matrix, we perform row operations until the matrix satisfies the following conditions:
1. The leftmost non-zero entry (leading entry) of each non-zero row is 1.
2. Each leading 1 is the only non-zero entry in its column.
3. All rows with all zero entries, if any, are at the bottom of the matrix.
Let's go step by step to find the reduced row echelon form of the given matrix:
Step 1: Start with the given matrix.
-1/2 1/2 3/2
-3/2 2 3/2
1/2 -3/2 9/2
Step 2: Perform row operations to make the leading entry of the first row equal to 1. Multiply the first row by -2.
1 -1 -3
-3/2 2 3/2
1/2 -3/2 9/2
Step 3: Perform row operations to make all entries below the leading 1 of the first row zero. Multiply the first row by 3/2 and add it to the second row. Multiply the first row by -1/2 and add it to the third row.
1 -1 -3
0 1/2 9/2
0 -1 15/2
Step 4: Perform row operations to make the leading entry of the second row equal to 1. Multiply the second row by 2.
1 -1 -3
0 1 9
0 -1 15/2
Step 5: Perform row operations to make all entries above and below the leading 1 of the second row zero. Multiply the second row by 1 and add it to the third row.
1 -1 -3
0 1 9
0 0 6.5
Step 6: Perform row operations to make the leading entry of the third row equal to 1. Multiply the third row by 1/6.5.
1 -1 -3
0 1 9
0 0 1
Step 7: Perform row operations to make all entries above the leading 1 of the third row zero. Multiply the third row by 1 and add it to the first row. Multiply the third row by -9 and add it to the second row.
1 -1 0
0 1 0
0 0 1
Step 8: Perform row operations to make all entries below the leading 1 of the third row zero. Multiply the third row by 3 and add it to the second row. Multiply the third row by 1 and add it to the first row.
1 0 0
0 1 0
0 0 1
Finally, we have achieved the row echelon form of the given matrix, and also the reduced row echelon form, as there are no entries above the leading 1 in any of the rows.
Therefore, the reduced row echelon form of the given matrix is:
1 0 0
0 1 0
0 0 1