[[4, 7], [2, 6]]
MATH / LINEAR ALGEBRA
2x2 Matrix Inverse Calculator
Calculate all four entries of an invertible two-by-two matrix inverse. Review the symbolic equation, substituted values, interpretation boundary, and result in one guided workspace.
- 01 Calculated in this tab
- 02 Values stay in this browser tab
- 03 Use boundary
ADJUGATE / IDENTITY CHECK
Turn four matrix entries into an inverse you can verify by multiplication
This explainer is designed for the part of a 2x2 inverse that usually gets split across several result rows. With the reviewed default matrix [[4, 7], [2, 6]], the calculator first finds det(A) = 10, then rescales the adjugate [[6, -7], [-2, 4]] by 1/10 to return A^-1 = [[0.6, -0.7], [-0.2, 0.4]]. The visual below keeps the swap-and-negate step and the identity check in one place.
Worked default example
- Entered matrix
- A = [[4, 7], [2, 6]], where a = 4, b = 7, c = 2, and d = 6.
- Invertibility check
- The calculator first checks det(A) = ad - bc. Here 4 x 6 - 7 x 2 = 10, so the matrix is invertible.
- Adjugate step
- Swap the diagonal entries and negate the off-diagonal entries to get adj(A) = [[6, -7], [-2, 4]].
- Scale every entry by 1 / det(A)
- 1/10 x 6 = 0.6; 1/10 x (-7) = -0.7; 1/10 x (-2) = -0.2; 1/10 x 4 = 0.4.
- Returned inverse matrix
- Read the main result and the three supporting metrics together: A^-1 = [[0.6, -0.7], [-0.2, 0.4]].
What to enter and how to read the result
Enter the four scalar entries in row order: top-left a, top-right b, bottom-left c, and bottom-right d. The main result card reports inverse entry (1,1), and the three metrics below it report entries (1,2), (2,1), and (2,2). Read all four together as the inverse matrix for the same 2x2 input.
What the calculator actually does
Nirmion evaluates A^-1 = (1 / (ad - bc))[[d, -b], [-c, a]] directly from the entered entries. In the reviewed default example, det(A) = 10, the adjugate is [[6, -7], [-2, 4]], and multiplying A by [[0.6, -0.7], [-0.2, 0.4]] reproduces the identity matrix [[1, 0], [0, 1]] through the four row-by-column checks shown in the audit table.
Where this model stops helping
This page handles real-number 2x2 entries only and requires a nonzero determinant. It does not keep symbolic fractions, invert larger matrices, diagnose near-singularity, or multiply the inverse by another matrix or constants vector for you. If det(A) = 0, the matrix is singular and has no inverse.
Reference and next checks
The linked OpenStax section states the 2x2 inverse formula and shows how an inverse is validated against the identity matrix. If you want to inspect the determinant first, solve a full two-equation system, or compare eigenvalues from the same matrix next, use the related calculators below.
Read OpenStax on solving systems with inverses2x2 inverse audit from original matrix to identity check
A four-card audit showing the original matrix [[4, 7], [2, 6]], the factor 1/10, the adjugate [[6, -7], [-2, 4]], the inverse [[0.6, -0.7], [-0.2, 0.4]], and the row-by-column checks that return the identity matrix [[1, 0], [0, 1]].
1/10 = 0.1
[[6, -7], [-2, 4]]
[[0.6, -0.7], [-0.2, 0.4]]
Identity product audit
Target product: [[1, 0], [0, 1]]
| Original matrix rows | Inverse column 1 | Inverse column 2 |
|---|---|---|
| A row 1 | Identity entry (1,1)4 x 0.6 + 7 x (-0.2)Result: 1 | Identity entry (1,2)4 x (-0.7) + 7 x 0.4Result: 0 |
| A row 2 | Identity entry (2,1)2 x 0.6 + 6 x (-0.2)Result: -0 | Identity entry (2,2)2 x (-0.7) + 6 x 0.4Result: 1 |
Continue the matrix workflow
Continue the matrix workflow
COMMON QUESTIONS
2x2 matrix inverse calculator FAQs
Why does the page show inverse entry (1,1) first instead of the whole inverse matrix?
The result panel uses one primary card and three supporting metrics. On this tool, that means the top-left inverse entry appears first, then entries (1,2), (2,1), and (2,2) appear underneath. Read the four values together as A^-1 = [[0.6, -0.7], [-0.2, 0.4]].
What changes when the determinant is negative?
A negative determinant does not block inversion by itself. It only changes the sign of the overall scaling factor 1 / (ad - bc). The matrix becomes non-invertible only when ad - bc equals zero.
Can I use this inverse to solve a two-equation system?
Yes. If your coefficient matrix is A and your constants vector is B, then the solution vector is X = A^-1 B. This calculator does not multiply by B for you, so use the related two-equation system calculator when you want x and y directly.
Use boundary
Calculation path
Calculate all four entries of an invertible two-by-two matrix inverse. The page evaluates the displayed equation from your supplied values and presents both symbolic and substituted KaTeX working so the arithmetic can be checked.
Calculation path
A^-1=(1/det A)[[d,-b],[-c,a]]
What you provide
What you provide
- Matrix a, using the same scenario as the other inputs
- Matrix b, using the same scenario as the other inputs
- Matrix c, using the same scenario as the other inputs
- Matrix d, using the same scenario as the other inputs
What you receive
What you receive
- Inverse entry (1,1) from the stated equation
- A symbolic formula plus substituted working with your values
- Visible assumptions, field guidance, and an authoritative learning reference
Use boundary
Choose the maximum decimal places shown. This does not increase source accuracy.
The determinant must be nonzero.
Use internally consistent units and retain extra precision when carrying the result into another calculation.
Reviewed reference factors