CALCULATOR GUIDE
REF stands for "row echelon form" meaning a matrix with a lower triangle of zeros. From this form you can use back-substitution to solve for the variables. But, if you are going to use a calculator to solve a system via matrix methods, why not go all the way to "reduced row echelon form" using RREF and be able to read off the answers immediately.
RREF can be performed on matrices entered using the home screen only or on those stored via the matrix editor feature.
Solving a System of Linear Equations using RREF:
| Solving a linear system using RREF on matrices entered in the home screen | Example 1. Solve the system: 4x + 3y - 6z = -14 2x - y + 4z = -1 5x - 2y + 2z = -7 Answer. x=-2, y=-1, z=½ |
| TI-84 This is just like the 89/92 (see below) except the rref command is in MATRIX MATH RREF and you need [ ] around the entire system and [ ] around each separate equation instead of using semi-colons. |
MATRIX MATH RREF This will bring up RREF( on the screen. You will fill it in as, RREF([[4,3,-6,-14][2,-1,4,-1][5,-2,2,-7]]) ENTER The answer will look sort of like this, |
| TI-86 2ND MATRX OPS RREF Your solutions can be read down the right side of the boxed answer you get. They go down in the order you entered the variables, x at the top, y next, z under that, etc. |
2ND MATRX OPS RREF This will bring up RREF on the screen. You will fill it in as, RREF [[4,3,-6,-14][2,-1,4,-1][5,-2,2,-7]] ENTER The answer will look sort of like this,
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| TI-89 2ND MATH MATRIX RREF Your solutions can be read down the right side of the boxed answer you get. They go down in the order you entered the variables, x at the top, y next, z under that, etc. |
2ND MATH MATRIX RREF This will bring up RREF( on the screen. You will fill it in as, RREF([4,3,-6,-14 ; 2,-1,4,-1 ; 5,-2,2,-7]) ENTER The answer will look sort of like this,
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| HP-48G << UNDER CONSTRUCTION >> |
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| Solving a linear system using RREF on matrices stored via the matrix editor feature | Example 1. Solve the system: 4x + 3y - 6z = -14 2x - y + 4z = -1 5x - 2y + 2z = -7 Answer. x=-2, y=-1, z=½ |
| TI-84 First enter the matrix and store it as A per the directions at "How to enter and store a matrix using the matrix editor feature". Now pull this matrix into the RREF command. |
2ND MATRIX MATH scroll down to RREF then ENTER This will display "rref(" on the screen. 2ND MATRIX NAMES The answer will look sort of like this, |
| TI-86 First enter the matrix and store it as A per the directions at "How to enter and store a matrix using the matrix editor feature". Now pull this matrix into the RREF command. |
2ND MATRIX OPS RREF ALPHA A This will display "rref A" on the screen. ENTER The answer will look sort of like this, |
| TI-89 First enter the matrix and store it as A per the directions at "How to enter and store a matrix using the matrix editor feature". Now pull this matrix into the RREF command. |
2ND MATH MATRIX MATH scroll down to RREF then ENTER This will display "rref(" on the screen. ALPHA A and close the parentheses ) ENTER The answer will look sort of like this, |
| HP-48G << UNDER CONSTRUCTION >> |
Example 2. Solve the following system on your calculator.
a - 2b - c + 3d = 12
2a + b - d - 3c = -5
-2a + 3b + 2c + d = -13
2b + c - 2d = -8Note:
- In the second equation be sure to rearrange the terms so they are in an order consistent with the other equations -> 2a + b - 3c - d = -5
- In the fourth equation there you will need to use a zero as a place holder where the a-term is missing -> 0a + 2b + c - 2d = -8So the matrix to be entered will be:
[ 1 -2 -1 3 12 ]
[ 2 1 -3 -1 -5 ]
[ -2 3 2 1 -13 ]
[ 0 2 1 -2 -8 ]The solution is a=3, b=-4, c=2, d=1. Did you get it? Now try solving this system algebraically instead and you will come to really appreciate your calculator!
Originally written: 2000-05-08
Last revision:
2008-11-18 01:21 PM
