Understanding Row Operations | How Different Sequences Can Lead to Multiple Reduced Echelon Form Matrices

In some cases, a matrix may be row reduced to more than one matrix in reduced echelon form, using different sequences of row operations.

Indeed, it is possible for a matrix to be row reduced to more than one matrix in reduced echelon form using different sequences of row operations

Indeed, it is possible for a matrix to be row reduced to more than one matrix in reduced echelon form using different sequences of row operations. This phenomenon occurs due to the flexibility and different possibilities offered by the row operations.

Row operations include three types:
1. Swapping two rows.
2. Multiplying a row by a non-zero scalar.
3. Adding a multiple of one row to another row.

When applying these operations, the goal is to reach the reduced echelon form, where every leading entry (the first non-zero entry) of each row is 1 and is the only non-zero entry in its column. Additionally, all the rows consisting entirely of zeros should be at the bottom of the matrix.

While there is a standard algorithm, known as Gaussian elimination, to row reduce a matrix, there are various valid sequences of row operations that can be performed. These different sequences can lead to different matrices in reduced echelon form that are equivalent to the original matrix.

To illustrate this, consider the following example:

Original matrix:
[ 1 2 ]
[ 3 6 ]

By applying different sequences of row operations, we can obtain the following result:

Sequence 1:
[ 1 2 ]
[ 0 0 ]

Sequence 2:
[ 0 1 ]
[ 1 3 ]

Both of these matrices are in reduced echelon form. Sequence 1 uses the operation of subtracting 3 times the first row from the second row to achieve the result, while Sequence 2 involves dividing the second row by 3 and then swapping the rows.

Therefore, it is crucial to note that there can be multiple sequences of row operations that can lead to different matrices in reduced echelon form, all of which represent the same solution space or set of solutions.

More Answers: