HOW TO REDUCE A MATRIX
Legal Row Operations:
1) Interchange 2 rows of the matrix
2) Multiple every entry in a row by a constant.
3) Multiply every entry in a row by a constant, then add that row to another row.
Note: Use operations 1)and 2) To create ones in a matrix. Use operation 3) to create zeros.
1 2 3
1 0 a 
Example 1. Reduce 
Your
goal
is
to
transform
it
to

0 1 b 
 4 5 6


Since there is already a 1 in the upper left corner, the next step is to create a zero where the 4 is.
Multiply every entry in row 1 by -4 and add those products to row 2.
3 
1 2
Now the matrix should look like this: 
 The next step is to create a 1 where the -3 is.
0 − 3 − 6
Step 2: Multiple each entry in row 2 by -1/3.
1 2 3
Now the matrix looks like this: 
 Finally, create a zero right above the 1 in row 2.
0 1 2
1 0 − 1
Step 3: Multiple every entry in row 2 by -2 and add those products to row 1 : 

0 1 2 
______________________________________________________________________________
 1 −2 3 9
1 0 0 a 


Example 2: Reduce  − 1 3 0 4 
Your goal is to transform it to 0 1 0 b 
 2 − 5 5 17 
0 0 1 c 
R1 + R2 ⇒ R2
− 2 R1 + R3 ⇒ R3
 1 −2 3 9
− 1 3 0 4  →


 2 − 5 5 17 
1 − 2 3 9 
0 1 3 13 →


 2 − 5 5 17
9
1 − 2 3
0 1
3 13  →

0 − 1 − 1 − 1
1 R3 ⇒ R3
2
1 0 9 35
0 1 3 13


0 0 2 12 
− 3 R3 + R2 ⇒ R2
− 9 R3 + R1 ⇒ R1
1 0 9 35
0 1 3 13 


0 0 1 6 
1 0 9 35 
 0 1 0 − 5


0 0 1 6 
→
2 R2 + R1 ⇒ R1
→
R2 + R3 ⇒ R3
→
9 35 
1 0
0 1
3 13 

0 − 1 − 1 − 1
1 0 0 − 19
0 1 0 − 5 


0 0 1
6 