RREF on TI-84 vs Online RREF Calculator

How to Compute RREF on a TI-84 Calculator

The TI-84 Plus, TI-84 Plus Silver Edition, and TI-84 Plus CE all include a built-in rref( function under the Matrix Math menu. Here is the complete step-by-step procedure:

1. Store your matrix. Press 2nd → MATRIX (or NAMES). Navigate right to EDIT. Select matrix [A]. Enter the number of rows and columns, then type each entry. Press 2nd → QUIT when done.
2. Open the Matrix Math menu. Press 2nd → MATRIX. Navigate right to MATH.
3. Select rref(. Scroll down to B: rref( (it may be labeled differently on older OS versions). Press ENTER.
4. Enter the matrix name. Press 2nd → MATRIX, select [A] from NAMES, press ENTER.
5. Close the parenthesis. Type ) and press ENTER.
6. Read the RREF. The result is displayed as a matrix. You may need to scroll right to see columns beyond the screen width.

For an augmented matrix [A|b], store all columns (including the b column) as a single matrix. For example, a 3-equation, 3-variable augmented matrix becomes a 3×4 matrix in [A].

The Floating-Point Problem on TI-84

The TI-84 stores all numbers in floating-point format with 14 significant decimal digits of precision. Most fractions — including 1/3, 1/7, 2/11, and many others — cannot be represented exactly in this format. They are rounded to the nearest representable value.

In a single arithmetic operation, this rounding error is negligible (roughly 10⁻¹⁴). But Gauss-Jordan elimination performs many operations, each using the results of previous ones. Errors accumulate. For a 3×3 matrix with fraction entries, the final RREF may contain values like:

For homework where you must show exact fraction answers, these floating-point residuals are problematic. A student who gets 9.9999e-13 cannot tell whether this is "effectively zero" or whether they made an error. The TI-84 has a Frac function (press MATH → 1: ►Frac) that attempts to convert decimals back to fractions, but it only works for simple fractions and fails for more complex cases.

How the Online Calculator Avoids This Problem

The RREF calculator on this site represents every number as an exact rational number: a pair of integers (numerator, denominator). The fraction 1/3 is stored as the pair (1, 3) — not as a decimal approximation. All arithmetic (addition, subtraction, multiplication, division) on these fraction pairs produces another exact fraction pair, following the standard rules for rational arithmetic.

This means there is no rounding at any stage. A value that should be zero is stored as the integer 0, not as 10⁻¹³. A value that should be 7/12 is stored as the pair (7, 12), not as 0.58333.... The output matches exactly what you would compute by hand with exact fraction arithmetic.

The tradeoff: the online calculator requires an internet connection and cannot be used on a closed-book exam. For exams and on-the-fly calculation, the TI-84 is the right tool — but for checking homework and understanding fraction results, the online calculator gives verifiably correct answers.

RREF on Casio Graphing Calculators

Casio calculators (fx-9750GIII, fx-9860GIII, fx-CG50) also include RREF functionality, accessed differently:

1. From the Main Menu, enter the Run-Matrix mode.
2. Press F1 (MAT/VCT) to open the matrix editor.
3. Enter your matrix dimensions and values, then press EXIT.
4. In the Run-Matrix editor, press OPTN → F2 (MAT) → F6 (more) → F4 (Rref).
5. Select your matrix and press EXE.

Like the TI-84, the Casio uses floating-point internally. The same rounding issues apply. Casio does have a fraction display mode (MathI/MathO settings), which helps for simple results, but does not eliminate the underlying floating-point arithmetic.

Side-by-Side Comparison

When to Use the TI-84

The TI-84 is the right choice when:

• Taking a closed-book exam that permits graphing calculators but not phones or laptops.
• Working with large matrices (more than 6 rows or columns) where the online calculator's size limit is reached.
• Doing quick approximate checks where floating-point results are close enough for your purpose.
• Chaining calculations using stored matrices in the calculator's memory for a sequence of operations.

When to Use the Online RREF Calculator

The online calculator is the right choice when:

• Your matrix contains fractions (1/3, 5/7, etc.) and you need exact fraction answers in the output.
• You need step-by-step explanations for each row operation — essential for learning and homework verification.
• You are checking your hand computation and need to know which specific step is wrong.
• You got a suspicious TI-84 result (like 9.9999e-13) and want to know if it should be zero.
• You need LaTeX output to paste into an assignment or paper.

For a complete worked tutorial using the online calculator, see our RREF step-by-step tutorial. For a comparison of Gauss-Jordan and Gaussian elimination methods, see the Gauss-Jordan vs Gaussian elimination guide.

Frequently Asked Questions