Question

Are the matrices A and B Row equivalent? Why or Why not?

A=

1	1	1
2	3	-1
-1	4	1

B =

1	0	-1
1	1	1
0	1	3

Answer 1

Two matrices are said to be row equivalent if one of these can be changed to the other by a sequence of elementary row operations or, alternatively if and only if the two matrices have the same row space.

To determine the row space of A, we will reduce it to its RREF as under:

Add -2 times the 1st row to the 2nd row

Add 1 times the 1st row to the 3rd row

Add 1 times the 1st row to the 3rd row

Add 1 times the 1st row to the 3rd row

Add 3 times the 3rd row to the 2nd row

Add -1 times the 3rd row to the 1st row

Add -1 times the 2nd row to the 1st row

Then the RREF of A is I₃.

Similarly, to determine the row space of A, we will reduce it to its RREF as under:

Add -1 times the 1st row to the 2nd row

Add -1 times the 2nd row to the 3rd row

Add -2 times the 3rd row to the 2nd row

Add 1 times the 3rd row to the 1st row

Then the RREF of B is I₃.

Hence { e₁,e₂,e₃} = {(1,0,0),(0,1,0),(0,0,1)] is the basis for the row space of A and also, the row space of B. Therefore, the matrices A and B are row equivalent.