Matrix Inverse Calculator
Calculate the inverse of 2x2 and 3x3 matrices with step-by-step solutions and determinant calculations
A⁻¹ = (1/det A) × adj(A)
Calculate the inverse of square matrices using the adjugate method
Quick Examples
Error
Determinant of Matrix A
Matrix is invertible ✓
Inverse Matrix A⁻¹
Adjugate Matrix adj(A)
Step-by-Step Solution
Calculate the determinant
det(A) =
Find the adjugate matrix
Swap diagonal elements, negate off-diagonal elements
Calculate cofactors and transpose the cofactor matrix
Calculate the inverse
A⁻¹ = (1/) × adj(A)
Verification
You can verify by multiplying A × A⁻¹ = I (identity matrix)
Matrix Inverse Formulas
| Size | Determinant | Inverse Formula |
|---|---|---|
| 2×2 | ad - bc | (1/det) × [[d,-b],[-c,a]] |
| 3×3 | Cofactor expansion | (1/det) × adj(A) |
If you like this calculator
Please help us simply by sharing it. It will help us a lot!
Related Calculators
Other calculators you might find useful.
Avogadro's Number Calculator
Convert between moles, atoms, and molecules using Avogadro's constant
Kinematics Calculator
Calculate motion parameters using the four kinematic equations for constant acceleration
Rational Zeros Calculator
Find all possible and actual rational zeros of a polynomial using the Rational Root Theorem
WHIP Calculator
Calculate Walks plus Hits per Innings Pitched for baseball pitchers
Goals Against Average Calculator
Calculate GAA for hockey goalies using goals allowed and minutes played
Cone Calculator
Calculate volume, surface area, slant height, and other properties of a cone
About Matrix Inverse Calculator
What is a Matrix Inverse?
The inverse of a matrix A (denoted A⁻¹) is a matrix that, when multiplied by A, yields the identity matrix:
A × A⁻¹ = A⁻¹ × A = I
Not all matrices have inverses. A matrix must be square (same number of rows and columns) and have a non-zero determinant to be invertible.
Formula for Matrix Inverse
The general formula for finding the inverse of a matrix is:
A⁻¹ = (1/det A) × adj(A)
Where:
- det A is the determinant of matrix A
- adj(A) is the adjugate (adjoint) of matrix A
2×2 Matrix Inverse
For a 2×2 matrix:
A = | a b |
| c d |
Determinant: det A = ad - bc
Inverse:
A⁻¹ = (1/(ad-bc)) × | d -b |
|-c a |
3×3 Matrix Inverse
For a 3×3 matrix, the process involves:
- Calculate the determinant
- Find the matrix of minors
- Apply the cofactor sign pattern
- Transpose to get the adjugate
- Multiply by 1/determinant
Applications
- Solving linear equations (Ax = b → x = A⁻¹b)
- Computer graphics (transformation matrices)
- Cryptography (Hill cipher)
- Engineering (circuit analysis, control systems)
- Economics (input-output models)
Limitations
- Singular matrices (det = 0) have no inverse
- Numerical instability with near-singular matrices
- Computational cost increases with matrix size