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) |
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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