Terry D. Johnson
10.001 Fall 2000
 
 
 

We begin with a general matrix of n by m dimensions:
 
 
 

Step 1: Focus on Row 1. Subtract a multiple of Row 1 from all rows beneath it. The multiple is chosen to eliminate a₂₁, a₃₁, a₄₁...a_m1. Example: (a₂₁/a₁₁)*Row 1 is subtracted from Row 2.

Note: When one row is subtracted from a second row, the numbers in the second row change. This change in number is denoted by the an accent (').
 
 
 

Step 2: Focus on Row 2. Subtract a multiple of Row 2 from all rows beneath it.  The multiple is chosen to eliminate a₃₂', a₄₂',...a_m2'. Example: (a₃₂/a₂₂)*Row 2 is subtracted from Row 3.
 
 
 

Step 3-Step (m-1): Continue until your matrix is in 'almost' Row Echelon form.
 
 
 

Back Substitute: Solve! Start at the bottom and solve for x_m = y_m'''^.../a_mn''''^...
 
 
 

then substitute x_m to solve for x_m-1. Use x_m and x_m-1 to solve for x_m-2.
 
 
 

~ Continue until x₁-x_m are solved for ~
 
 

END