What makes 2 equations row equivalent?
If there is a series of elementary row operations that transforms one matrix into the other
Two equations are said to be row equivalent if they have the same solution set. In other words, if one equation can be transformed into another by a sequence of operations on rows of its corresponding augmented matrix, then the two equations are row equivalent.
There are three elementary row operations that can be performed on a matrix that will not change the solution set of the equations represented by the matrix. These operations are:
1. Interchanging two rows
2. Multiplying a row by a nonzero constant
3. Adding a multiple of one row to another row
Using these operations, we can manipulate the matrix to obtain an equivalent matrix that represents an equivalent system of equations. If two matrices can be transformed into each other by a sequence of these operations, then they are said to be row equivalent, and the corresponding systems of equations have the same solution set.
As an example, consider the following two systems of equations:
System 1:
x + 2y = 7
3x – y = 1
System 2:
3x – y = 1
x + 2y = 7
We can see that the two systems are row equivalent by performing the following operations on the augmented matrices:
1. Swap rows 1 and 2 of System 1 to obtain System 2
2. Multiply row 1 of System 2 by -1/3 to obtain:
– x + 1/3y = -1/3
x + 2y = 7
3. Add row 1 of System 2 to row 2 to obtain:
– x + 1/3y = -1/3
7/3y = 20/3
4. Multiply row 2 of System 2 by 3/7 to obtain:
– x + 1/3y = -1/3
y = 20/7
Thus, we have transformed System 1 into the equivalent system:
– x + 1/3y = -1/3
y = 20/7
which has the same solution set as System 2. Therefore, System 1 and System 2 are row equivalent.
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