Matrix Inverse Calculator (2×2)
Invert a 2×2 matrix and solve the linear system.
Enter your values
Enter the a (row 1, col 1) you are working with.
Enter the b (row 1, col 2) you are working with.
Enter the c (row 2, col 1) you are working with.
Enter the d (row 2, col 2) you are working with.
Enter the right-hand side 1 (optional) you are working with.
Enter the right-hand side 2 (optional) you are working with.
Results update instantly as you type — no submit needed. Your values are remembered on this device, and the shareable link reopens the calculator with exactly these numbers.
- Inverse row 1
- [0.6, -0.7]
- Inverse row 2
- [-0.2, 0.4]
- Determinant
- 10
- Solution x
- -0.4
- Solution y
- 2.8
Quick answer
The inverse of [[a, b], [c, d]] is 1/(ad − bc) × [[d, −b], [−c, a]], which exists only when the determinant ad − bc is non-zero.
A⁻¹ = 1/det × [[d, −b], [−c, a]]
At a glance
| What it does | Invert a 2×2 matrix and solve the linear system. |
|---|---|
| Category | Math |
| Inputs needed | a (row 1, col 1), b (row 1, col 2), c (row 2, col 1), d (row 2, col 2), Right-hand side 1 (optional), Right-hand side 2 (optional) |
| Main output | Inverse row 1 |
| Formula | A⁻¹ = 1/det × [[d, −b], [−c, a]] |
| Cost | Free — no sign-up, no download |
How to use the Matrix Inverse Calculator (2×2)
- 1Enter the four matrix entries reading left to right, top to bottom.
- 2Read the inverse matrix.
- 3Optionally enter the right-hand side of your system to get x and y directly.
Inputs explained
- a (row 1, col 1)
- Enter the a (row 1, col 1) you are working with.
- b (row 1, col 2)
- Enter the b (row 1, col 2) you are working with.
- c (row 2, col 1)
- Enter the c (row 2, col 1) you are working with.
- d (row 2, col 2)
- Enter the d (row 2, col 2) you are working with.
- Right-hand side 1 (optional)
- Enter the right-hand side 1 (optional) you are working with.
- Right-hand side 2 (optional)
- Enter the right-hand side 2 (optional) you are working with.
Worked example
Using the values the calculator loads with:
Inputs
- a (row 1, col 1)4
- b (row 1, col 2)7
- c (row 2, col 1)2
- d (row 2, col 2)6
- Right-hand side 1 (optional)18
- Right-hand side 2 (optional)16
Results
- Inverse row 1[0.6, -0.7]
- Inverse row 2[-0.2, 0.4]
- Determinant10
- Solution x-0.4
- Solution y2.8
Frequently asked questions
Why must the determinant be non-zero?
Dividing by the determinant is the last step of the formula; a zero determinant means the matrix collapses two dimensions into one and cannot be undone.
Can I check the answer?
Multiply the original by the inverse — you should get the identity matrix [[1,0],[0,1]].
What do I need to enter into the Matrix Inverse Calculator (2×2)?
Just 6 values: a (row 1, col 1), b (row 1, col 2), c (row 2, col 1), d (row 2, col 2), right-hand side 1 (optional), right-hand side 2 (optional). Every field starts with a realistic example, so you can change one number at a time and watch the result update instantly.
How does the Matrix Inverse Calculator (2×2) work out the answer?
The inverse of [[a, b], [c, d]] is 1/(ad − bc) × [[d, −b], [−c, a]], which exists only when the determinant ad − bc is non-zero. It applies the formula A⁻¹ = 1/det × [[d, −b], [−c, a]] and shows the working so you can check each step by hand.
Is the Matrix Inverse Calculator (2×2) free, and do I need an account?
It is completely free with no sign-up, no download and no usage limit. Everything is calculated in your browser, so the numbers you type never leave your device.
Results are estimates for general information. For medical, legal, structural or financial decisions, confirm with a qualified professional.