SOLUTION: Prove or disprove whether the following matrices are equivalent or not. Explain how you came up with your answer. 123 011 325 549

Prove or disprove whether the following matrices are 
equivalent or not. Explain how you came up with your answer. 

[1 2 3]
[0 1 1] 

[3 2 5]
[5 4 9]

We must get both of these into "row reduced echelon form"

Let's get the first one to row reduced echelon form:

[1 2 3]
[0 1 1]

Get a 0 where the 2 is by multiplying the bottom row by -2
and adding it to 1 times the top row, then restoring 
the bottom row:

 1[1 2 3]
-2[0 1 1]

[1 0 1]
[0 1 1]

That is in row reduced echelon form because:
1. the leading 1 in the bottom row is farther
   to the right than the leading 1 in the top
   row.
2. Each of the leading 1's are the only non-zero
   numbers in the columns they are in. 

Now, let's get the second one to row reduced echelon form:

[3 2 5]
[5 4 9]

Get a 0 where the 5 on the bottom row is by multiplying the 
top row by -5 and adding it to 3 times the bottom row, then 
restoring the top row:

-5[3 2 5]
 3[5 4 9]

[3 2 5]
[0 2 2]

Get a 0 where the 2 on the top row is by multiplying the 
bottom row by -1 and adding it to 1 times the top row, then 
restoring the bottom row:

 1[3 2 5]
-1[0 2 2]

[3 0 3]
[0 2 2]

Get a 1 where the first 3 in the top row is by dividing the
top row through by 3.

Get a 1 where the first 2 in the bottom row is by dividing the
bottom row through by 2.

[3 0 3]÷3
[0 2 2]÷2

[1 0 1]
[0 1 1]

So the given matrices are equivalent because the have the
same row reduced echelon form, namely

[1 0 1]
[0 1 1] 

Note: It didn't apply here but in some matrices, to have 
row reduced echelon form, all all-zero rows must appear
at the bottom of the matrix.  This can be accomplished
by swapping rows.

Edwin