The inverse undoes a matrix: A⁻¹A = I. It solves systems of equations (x = A⁻¹b), reverses transformations in graphics and appears throughout statistics and engineering.
How to use the Inverse Matrix Calculator
- Choose the matrix size – the matching grid of fields appears.
- Enter the entries row by row (a₁ b₁ c₁ is the first row, a₂ b₂ c₂ the second, and so on).
- Read the result; integer matrices give exact fractions.
Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.
How it works
The calculator computes the determinant and the adjugate; dividing gives the inverse. A matrix with determinant 0 is singular and has no inverse.
Formula
A⁻¹ = adj(A) / det(A)
2 × 2: [[a, b], [c, d]]⁻¹ = 1/(ad − bc) · [[d, −b], [−c, a]]
Worked example
Inputs — Matrix size: 2 × 2; a₁: 4; b₁: 7; a₂: 2; b₂: 6.
Results — Inverse A⁻¹ = ⎡ 3/5 −7/10 ⎤
⎣ −1/5 2/5 ⎦; Determinant |A| = 10; Check = A × A⁻¹ = I ✓; A⁻¹ as decimals = ⎡ 0.6 −0.7 ⎤
⎣ −0.2 0.4 ⎦.
Key terms
- Singular matrix
- A square matrix with determinant 0.
Frequently asked questions
When does a matrix have an inverse?
When its determinant is non-zero.
What is the inverse of [[4, 7], [2, 6]]?
1/10 · [[6, −7], [−2, 4]].
Is (AB)⁻¹ = A⁻¹B⁻¹?
No – (AB)⁻¹ = B⁻¹A⁻¹.
Should I invert a matrix to solve Ax = b?
For hand work yes; numerically, elimination is faster and more accurate.