Press MATRX, select EDIT (F2),  The "A" seen under the cursor indicates that alphabetic mode is on.
Name your matrix:  A
ENTER
Define the dimensions of the matrix.  2 ENTER 3 ENTER
Enter each matrix element.  After typing each element, press ENTER.
This defines the matrix [A]. QUIT to return to the home screen.

We will do a series of elementary row operations on the matrix [A] defined above to place it in row-echelon form (by forward elimination) and then reduced row-echelon form (by backward elimination).  We assume [A] has already been defined, as shown above.
 

Forward Elimination
To see our matrix A:  MATRX, select EDIT (F2), A ENTER
Go to matrix operations:  QUIT MATRX select OPS (F4)
Row operation:  Get a 1 in upper left-hand corner byR1ßà R2
MORE to move the menu to the right, then select rSwap (F2)
select matrix A:  select NAMES (M1) and A (F1)
, 1, 2) ENTER You now see matrix A with its rows swapped.
Save (store) this matrix back in A to prepare for the next operation.  STO A ENTER
Row operation:  Make the 3 a 0 by -3R1 + R2à R2
select OPS (M4), select mRAdd (F5)
-3, select NAMES (M1) select A (F1)  , 1, 2) ENTER
STO A ENTER
Row operation:  Make the 10 a 1 by (1/10)R2à R2
select OPS (M4), select multR (MOREF4)  .1, ALPHA A, 2) ENTER STO A ENTER
The matrix is now in row-echelon form.
Backward Elimination
Row operation:  Make the -2 (in row 1) a 1 by 2R2 + R1à R1
select OPS (M4), select mRAdd (MORE F5)  2, ALPHA A, 2, 1) STO A ENTER
The matrix is now in reduced row-echelon form, and the answer is: x = 3, y = -2

This operation performs forward elimination (see Row operations, above) on a matrix in one step.
 

Set FLOAT mode to 3 decimal places of precision: MODE down arrow to select FLOAT, right arrow to select 3, ENTER
We start with the same matrix A as above (now with 0's as specified in step 1):  ALPHA A ENTER
Go to matrix ops  MATRX F4
Select "ref", specify A  F4 ALPHA A ENTER
Our matrix is now in row-echelon form.  Note that it is not the same as obtained when we did the reduction steps by hand above.  This is because row-echelon form is not unique.

 

This operation does forward and backward elimination (see Row operations, above) in one step.
 

Starting with matrix A (see Row-echelon form for setting precision). ALPHA A ENTER
Go to matrix ops  MATRX F4
Select "rref", specify A  F5 ALPHA A ENTER
Our matrix is now in reduced row-echelon form.  Notice that this formis the same as found when we did the steps one by one.