Working out the inverse of a 3×3 matrix by hand can be time-consuming. Luckily, most modern calculators—like the TI-84 graphing calculator, Casio fx-991EX scientific calculator, and HP Prime—make it much easier. With the right steps, you can quickly enter the matrix, press the x⁻¹ key, and get the result.
In this guide, we’ll show you exactly how to calculate the inverse of a 3×3 matrix on different types of calculators, what to do if your calculator shows an error, and how to verify your results. For double-checking, you can also use the free online Inverse Matrix Calculator.
Key Things to Know Before You Start
- Only square matrices (same number of rows and columns) can be inverted.
- A matrix is invertible if its determinant ≠ 0.
- The result should return the identity matrix (I₃) when multiplied by the original.
- Some calculators show results in decimals, while others allow fraction form.
How to Invert a 3×3 Matrix on a TI-84 or TI-83
- Press [2nd] → [MATRIX].
- Scroll to EDIT, choose matrix
[A]. - Set dimensions to 3×3.
- Enter the numbers row by row.
- Exit to the home screen.
- Press [2nd] → [MATRIX] → [NAMES], select
[A]. - Press the x⁻¹ key.
- Hit ENTER → The inverse appears.
✅ Tip: If the calculator says “ERROR: SINGULAR MATRIX”, the determinant is zero, so the inverse doesn’t exist.
How to Invert a 3×3 Matrix on Casio Scientific Calculators (fx-991ES / fx-991EX / fx-115ES)
- Press MODE and select Matrix (MAT).
- Choose MatA.
- Set dimensions to 3×3.
- Input all elements row by row.
- Exit input mode.
- Type
MatA, then press the x⁻¹ key. - Press = → The inverse matrix appears.
💡 Casio calculators allow switching between fractions and decimals with the S⇔D key.
How to Invert a 3×3 Matrix on HP Calculators
- Open the Matrix Editor.
- Define a 3×3 matrix.
- Enter the elements.
- On the home screen, select the matrix and press x⁻¹.
- The calculator displays the inverse.
Worked Examples
Example 1: Integer Matrix
A=[210−1342−21]A = \begin{bmatrix} 2 & 1 & 0 \\ -1 & 3 & 4 \\ 2 & -2 & 1 \end{bmatrix}
After inputting into your calculator, pressing x⁻¹ will give its inverse.
Example 2: Singular Matrix
B=[123456789]B = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}
This matrix has determinant = 0. Your calculator will show error because it has no inverse.
Troubleshooting Common Issues
- Error: Singular Matrix → Determinant = 0, no inverse exists.
- Wrong dimensions → Make sure you selected 3×3, not 2×2.
- Decimal vs fraction output → Use calculator settings or toggle S⇔D.
- Memory limitations → Some calculators can only handle up to 3×3 or 4×4.
Comparison of Calculator Models
| Calculator | Key Sequence for 3×3 Inverse | Max Matrix Size | Output |
|---|---|---|---|
| TI-84 / TI-83 | [2nd] → [MATRIX] → EDIT → x⁻¹ | Up to 10×10 | Decimal & Fraction (depending on mode) |
| Casio fx-991ES/EX | MODE → Matrix → MatA → x⁻¹ | Up to 3×3 | Fraction & Decimal toggle |
| HP Prime / HP 50g | Matrix Editor → x⁻¹ | Large (depends on model) | Decimal by default |
Verifying the Inverse
To confirm your calculator’s result:
- Multiply the original matrix by its inverse.
- The result should be the 3×3 identity matrix:
I3=[100010001]I₃ = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}
- Alternatively, cross-check with the free Inverse Matrix Calculator online.
FAQs
Q1: Can all calculators find 3×3 inverses?
No, only models with matrix support like TI-84, Casio fx-991ES/EX, or HP Prime.
Q2: Why does my calculator show “Error: Singular Matrix”?
The determinant is 0, meaning the matrix has no inverse.
Q3: Can calculators show exact fractions?
Yes—Casio and some TI models allow fraction mode.
Q4: How do I know if my matrix is invertible?
Check the determinant. If it’s ≠ 0, the inverse exists.
Q5: What’s the fastest way to verify results?
Multiply the original × inverse (should equal identity), or use an online calculator.