How to Find Inverse of 3×3 Matrix on Calculator – Step-by-Step Guide

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

1. Press [2nd] → [MATRIX].
2. Scroll to EDIT, choose matrix [A].
3. Set dimensions to 3×3.
4. Enter the numbers row by row.
5. Exit to the home screen.
6. Press [2nd] → [MATRIX] → [NAMES], select [A].
7. Press the x⁻¹ key.
8. 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)

1. Press MODE and select Matrix (MAT).
2. Choose MatA.
3. Set dimensions to 3×3.
4. Input all elements row by row.
5. Exit input mode.
6. Type MatA, then press the x⁻¹ key.
7. 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

1. Open the Matrix Editor.
2. Define a 3×3 matrix.
3. Enter the elements.
4. On the home screen, select the matrix and press x⁻¹.
5. 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:

1. Multiply the original matrix by its inverse.
2. 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}

3. 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.