Key takeaways
- A⁻¹ = (1 ÷ (ad − bc)) × [d −b; −c a].
- Swap a and d, change the signs of b and c, then divide everything by the determinant.
- If ad − bc = 0, the matrix is singular and has no inverse.
- Check: A × A⁻¹ must equal [1 0; 0 1].
In this guide
What is a matrix inverse?
For numbers, 5 × (1/5) = 1. For matrices, the inverse A⁻¹ is the matrix that gives the identity matrix when multiplied by A. So A × A⁻¹ = I.
If A scrambles something, A⁻¹ unscrambles it.
Why does the inverse matter?
The inverse lets you solve matrix equations. If A × x = b, then x = A⁻¹ × b, just as 5x = 20 gives x = 20 ÷ 5. This is how computers solve systems of equations, undo transformations and decode messages.
Not every matrix can be undone. If the determinant is 0, the matrix has squashed information flat, and nothing can bring it back.
The formula
| Symbol | What it means | Example |
|---|---|---|
| a, b, c, d | Entries of A = [a b; c d] | 2, 1, 5, 3 |
| ad − bc | The determinant, must not be 0 | 1 |
| [d −b; −c a] | Swap a and d, flip signs of b and c | [3 −1; −5 2] |
| A⁻¹ | The inverse matrix | [3 −1; −5 2] |
The calculator shows the full inverse matrix in its steps. The main result is the top-left entry, d ÷ det. If the determinant is 0, it tells you the matrix is singular.
How to find the inverse step by step
- 1Find the determinant.
ad − bc. If it is 0, stop: there is no inverse. - 2Swap a and d. They trade places on the main diagonal.
- 3Change the signs of b and c. They stay in place, but positive becomes negative and negative becomes positive.
- 4Divide every entry by the determinant.
- 5Check. Multiply A by your answer. You should get [1 0; 0 1].
Enter your own matrix in the calculator below to see the full inverse.
Try it yourself
Pre-filled with the example — change any value.
Common mistakes to avoid
- Forgetting to divide by the determinant. The swapped matrix alone is not the inverse unless the determinant is 1.
- Sign errors. If b is −3, then −b is +3.
- Trying to invert a singular matrix. Always check the determinant first.
Tips and tricks
- Determinant of 1? Then the inverse is just the swap-and-flip matrix, with no fractions.
- Keep fractions if the determinant does not divide neatly, for example 1/3 instead of 0.333.
- Inverse of the inverse is the original matrix: (A⁻¹)⁻¹ = A.
Where you'll use it in real life
- Algebra: solving two simultaneous equations with x = A⁻¹b.
- Graphics: undoing a rotation or stretch.
- Codes: decoding messages that were scrambled with a matrix.
- Economics: input–output models that link different industries.
Quick summary
To invert a 2×2 matrix, find the determinant, swap a and d, change the signs of b and c, and divide by the determinant. If the determinant is 0, there is no inverse. Use the calculator above to get the full inverse with every step shown.
Worked example
A simple secret code scrambles pairs of numbers using the matrix [2 1; 5 3]. To decode messages, you need the inverse matrix.
- 01det = ad - bc
- 02det = (2)(3) - (1)(5) = 1
- 03A⁻¹ = (1/1) × [3 -1; -5 2]
- 04A⁻¹ = [3 -1; -5 2]
These numbers are pre-filled in the calculator above.
Frequently asked questions
When does a 2×2 matrix have no inverse?
When its determinant ad − bc equals 0. Such a matrix is called singular.
How do I check if my inverse is correct?
Multiply the original matrix by your inverse. If the answer is [1 0; 0 1], the inverse is correct.
What is the inverse of [2 1; 5 3]?
The determinant is 2 × 3 − 1 × 5 = 1, so the inverse is simply [3 −1; −5 2].
