Given the initial augmented matrix
-1 1 2 0
2 -1 -3 0
2 -2 -4 0 find the RREF
To find the Reduced Row Echelon Form (RREF) of the given augmented matrix, we can apply the Gauss-Jordan elimination method. The goal is to transform the given matrix into a form such that all the leading coefficients (the leftmost non-zero entry in each row) are equal to 1, and each leading coefficient is the only non-zero entry in its column. The RREF will also have zeros above and below each leading coefficient.
Let's perform the row operations step by step:
Step 1: Swap rows if necessary to bring a non-zero entry on the top-left corner of the matrix.
In this case, the top-left entry is already non-zero, so no need to swap rows.
Step 2: Scale the first row to make its leading coefficient equal to 1.
Divide the first row by -1:
1 -1 -2 0
Step 3: Eliminate the values below the leading coefficient of the first row.
Multiply the first row by 2 and add it to the second row:
1 -1 -2 0
0 0 -7 0
Multiply the first row by 2 and add it to the third row:
1 -1 -2 0
0 0 -7 0
0 0 0 0
Step 4: Scale the second row to make its leading coefficient equal to 1.
Divide the second row by -7:
1 -1 -2 0
0 0 1 0
Step 5: Eliminate the value above the leading coefficient of the second row.
Multiply the second row by -2 and add it to the first row:
1 -1 0 0
0 0 1 0
Step 6: The matrix is now in RREF:
1 -1 0 0
0 0 1 0
0 0 0 0
Now we have the RREF of the given augmented matrix.